There is no problem with installing junction boxes above a suspended ceiling, as long as the box is less than 100 in.³ and securely fastened. National Electrical Code has this to say...

National Electrical Code 2014

Chapter 3 Wiring Methods and Materials

Article 314 Outlet, Device, Pull, and Junction Boxes; Conduit Bodies; Fittings; and Handhole Enclosures

II. Installation

314.23 Supports. Enclosures within the scope of this article shall be supported in accordance with one or more of the provisions in 314.23(A) through (H).

(D) Suspended Ceilings. An enclosure mounted to structural or supporting elements of a suspended ceiling shall be not more than 1650 cm3 (100 in.3) in size and shall be securely fastened in place in accordance with either (D)(1) or (D)(2).

(1) Framing Members. An enclosure shall be fastened to the framing members by mechanical means such as bolts, screws, or rivets, or by the use of clips or other securing means identified for use with the type of ceiling framing member(s) and enclosure(s) employed. The framing members shall be adequately supported and securely fastened to each other and to the building structure.

(2) Support Wires. The installation shall comply with the provisions of 300.11(A). The enclosure shall be secured, using methods identified for the purpose, to ceiling support wire(s), including any additional support wire(s) installed for that purpose. Support wire(s) used for enclosure support shall be fastened at each end so as to be taut within the ceiling cavity.

Code Violations

In the first image, the wires being spliced together like that violates NEC 300.15.

300.15 Boxes, Conduit Bodies, or Fittings Where Required. ... Where the wiring method is conduit, tubing, Type AC cable, Type MC cable, Type MI cable, nonmetallic-sheathed cable, or other cables, a box or conduit body shall be installed at each conductor splice point, outlet point, switch point, junction point, termination point, or pull point...

I've also noted some violations in the second image.

^{Click for larger view}

Let's see if I can draw this all once again...

CAVEAT: I'm an electrical engineer, not an electrician. My description of the circuits will be correct, but someone should double-check me on the color-coding conventions.

The basic concept of a three-way circuit is that you have two single-pole, double-throw switches hooked up back to back:

        Switch 1                             Switch 2     
Black     .___________Black Traveler_____________.           Black
Hot ____./                                         .________ Switched Hot to load
          .___________Red   Traveler_____________./
White                                                        White  
Neutral_______________White  Neutral________________________ Neutral to load

Green or bare                                               Green or bare 
Ground _______________Green/bare ground____________________ Ground to load

When both Switch 1 and Switch 2 select the Black Traveler, or both select the Red Traveler, current flows and the load is powered. When they don't agree (one selects BlackT and the other selects RedT), current can't flow and the load gets no power.

The most common error made when connecting these is to get one of the switches "backward", connecting its common to one of the travelers. This results in a circuit that looks like:

        Switch 1                             Switch 2     
          .___________Black Traveler__________________
Hot ____./                                         .__)
          .___________Red   Traveler_____________.              Black Switched Hot
                                                  \._________   To Load

White                                                           White  
Neutral _______________White  Neutral________________________   Neutral to load


Green or bare                                               Green or bare 
Ground _______________Green/bare ground____________________ Ground to load

Now, if Switch 1 is up (BlackT), then it doesn't matter which way Switch 2 is set, since there's no way current can get from the top of Switch 2 to the bottom (they can't both be selected at once) and the load gets no power. If Switch 1 is down (RedT), and Switch 2 is up (BlackT), both travelers are now hot, but there's still no power to the load. Only if both switches are down, connecting hot to RedT and RedT to the load, will current flow.

"Oops."

The fix, of course, is to figure out which switch is hooked up incorrectly and turn it around so it faces the right way. If the circuit has been properly color-coded (in the US, that means the travelers are black and red; it doesn't much matter which is which), you can tell this by inspection; each switch should have a black wire connected to its common, and a black and red (from the same cable) connected to its other two terminals. So if you can determine which of the two cables the red belongs to, you can see which switch is hooked up incorrectly and fix it immediately.

If you can't figure out which black belongs to the same cable that the red does, in either box, don't panic. Assuming that both switches do have a red and a black at their outputs, what has almost certainly happened is that the installer picked the wrong black for at least one of the switches. Swap the blacks at one switch (you DID turn off the breaker before opening the switch box, right?!?), turn the breaker back on and see if the switches now work as intended. If not, turn the breaker back off and swap the blacks at the other switch; turn the breaker back on and try again. If it STILL doesn't work as expected, turn the breaker off one more time, go back to the first switch and put the blacks back the way they were; this is the only remaining possibility and when you turn the breaker on one last time it should (finally!) work.

If it still doesn't work as intended, you've got a malfunctioning switch or bad wiring, and you should probably go after it with a multimeter or voltage probe to figure out which. Of course you could have used a probe earlier and saved yourself all the experimentation, but I've been presuming all that's available is a screwdriver and patience and the Mark 1.0 Eyeball.

Note: If you don't see the expected color coding -- one cable having black and white wires, and the other having black, white, AND RED wires -- STOP and find out what your country's conventions are for this, and/or figure out how the weirdo who set this up actually wired it.

There's one other complication, though.

I've been describing the most common setup, where the load is connected to the second switchbox. You can confirm that by the fact that there are two cables coming into each box, each switch has a black at common and a black and red as the two travellers, and the two whites (neutrals) in each box are connected to each other (so neutral runs alongside the travelers and is available in the second box for connection to the load).

Sometimes, however -- typically when the second switch was added later -- the load is actually connected at the first switchbox rather than the second. In this case, what you'll see is THREE cables coming into one box (Switch A) and ONE cable coming into the second box (Switch B). This winds up looking like:

        Switch 1                              Switch 2     
Black     .___________Black Traveler_____________.
Hot ____./                                         .________ 
          .___________Red   Traveler_____________./          \ 
White                                                         )
Neutral____      _____Return Traveler_______________________ /
           |    |
      White     Black
      Neutral   Switched hot
          TO LOAD
           Ground  
Ground ______|________Green/bare ground____________________ Ground

OK, that works... but now we need to figure out what color the Return Traveler should be. The US-standard three-wire-plus-ground cable only gives us the four colors we've mentioned: Black for hot, White for neutral, Red for "alternate hot", and Green for safety ground. We don't want to have to buy another kind of cable just for this uncommon usage...

So there is, essentially, a Standard Cheat. In this setup, the Return Traveler uses the white wire in the traveler cable, BUT the electrician is Morally Obligated, On His Honor As A Member Of The Guild, to tag it with red electrical tape (or hit it with a red marker, at least!) at both ends so the next guy to open the box knows that it is intended to be a Second Alternate Hot -- another "red" -- rather than a neutral.

(Of course gods only know whether homeowner-installed circuits have been properly labelled this way. You can hope that the guy who was there knew what he was doing and left you this indication, but don't bet your life on it!)

The same problem of a reverse switch (or two) can occur in this case, of course. But now it should be a bit more obvious how to correctly hook up Switch Two, since the common should be this Second Alternate Hot... so if one of the switches is backward it almost has to be Switch One. And if that isn't the cause, as above, start checking that the switches actually work and that the wiring is intact.

Whew.

There will be a quiz next Thursday. If you have any questions, please ask your TA during the recitation section. I'm gonna go get a cold bubbly.

(One advantage us old farts have: We still remember how to draw ASCII graphics!)