Often with light fixtures, each light has a pair of wires that come up and out of the fixture. You wire all of the hots together with the line hot, and all of neutrals together with the line neutral. If the wire nuts that connect all of the wires were not installed properly, it is possible that one of the wires is no longer making a solid electrical connection. It is also possible that one of the wires is damaged.
I would suggest that you remove the light fixture and examine all of the connections. Look for loose wires (you should not be able to easily pull them out of the wirenut), nicks, burns, etc.
If everything is visually OK you should test each wire to each socket with a multimeter. Test for continuity by ensuring resistance goes close to 0 when you make a connection. Repeat this until you find the problem. Make sure the power is off when you do this.
The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
Best Answer
Any time I see a strange voltage like 40 volts, I immediately suspect a broken wire or a bad terminal. Since installing and removing one of the bulbs seems to cause a problem, I would start by removing that fixture and inspecting it and the nearby wiring for faults.