You can't get rid of the telephone box in the middle. The FIOS connection provides telephone access to the FIOS panel and then the lines running from the FIOS panel to the telephone panel are used to connect all of the phones in your home to the FIOS telephone connection.

As for what you can do to make the situation better, I might plant a small tree, maybe a dogwood, that still leaves the window exposed below for security. I would also replace all the connections that are going in and around that window with a single 1-2" conduit and reroute all the lines through that (the phone wires going into the top of the window really bother me). And finally, I would get that box (electrical outlet?) over the vent properly attached.

Since all the telephone companies were deregulated, the practice is to have a demarcation point (which I'm really surprised no one else has mentioned yet). Most homes built within the last 20 years or so will have one, which will be a box or jack (often marked "DEMARC") marking the change in responsibility between their network and your in-house wiring.

Presumably, being a DIY site, you'd like to avoid being charged $85+ for something you can fix yourself, so there are really two steps to take:

• Isolate the problem to either their equipment or something before the demarc, or your in-house wiring
• If after demarc, isolate the problem to a particular jack

Connect your modem to the demarc

Best thing to do is disconnect all your house wiring from the demarc (usually it plugs in) and connect your modem to it, then run speed tests or whatever else. You may also want to try a different cable just to eliminate that.

Plug your PC/laptop directly into the modem if you can, to eliminate router/wireless, or at least plug your PC directly into to the router.

If you still have the problem, then it's either a problem with the wiring before the demarc, or your modem (both of which are the company's responsibility), or something on your PC/router (which they'll charge you for, and off-topic for this site).

If your Demarc is outside, you'll have to be a bit more creative, but at least you can just have one jack connected at a time (see the in-house wiring bit below).

Create a Demarc

If you have a wire that comes in and is directly hardwired to the house wiring, I'd suggest cutting it off and putting a demarc jack in (like the second picture).

Use a second jack to hardwire the rest of your house to, then a regular station cord to patch between the two. This also lets you disconnect the patch cable in order to do work on in-house wiring without worrying about shorting anything out or shocking yourself.

In-house wiring

If it's working at the demarc, the you've isolated the problem to the house. This could be much harder to diagnose.

If you have the same number of wires as jacks, then you have centralized wiring. If not, you probably have a harder-to-diagnose mess.

The new way: Star topology

If you're lucky, each jack will have its own cord coming back to a central location, where they'll probably all be connected together. Detach all but the one you're using (this may take some trial and error) and try the modem there. If it works, then there's a fault in another wire and you can isolate that one-by-one, or just leave it disconnected. Repeat for another jack or two in the house.

The old way: Branching/Daisy-chaining

Technically for phones to work, the ring and tip wires just need to be connected in parallel. Many times the jacks are all or partially daisy-chained, or even worse, there are junction points hidden in the walls, in whatever way works to make the shortest wiring possible. One of the problems with this is if there's a fault, it's extremely hard to find.

You pretty much have to follow the same steps -- trying to isolate lines -- but you'll have to disconnect wiring in the boxes around the house (and hope there are no hidden junctions). It may be simpler to just run a new jack and new wiring from the demarc to the spot you need, and leave everything else disconnected.