If you are not using encryption -- password protection -- then neighbors may be leeching onto your connection. I disagree with the person who implied that if you are using WEP this is a likely cause because a neighbor has broken your passkey -- it is a highly unlikely cause. Yes, WEP is breakable, but that's different from saying that it's frequently broken. Nevertheless, you really should be using WPA; it too is breakable but you've got to be REALLY good, like working for the NSA or something, to break it. But again, if you ARE password protected, even with WEP, this probably isn't your problem.
The location of your router in your new apartment is one likely cause of a slowdown. If
there are walls between your router and the place where you normally use your PC, and if those walls are
reinforced with a metallic mesh (like some load-bearing walls), this
could cause a major slowdown.
The farther your computer is from your router, and the more material (walls, furniture, floors, etc.) it has to pass through, the slower and less reliable your connection will be.
You can test this easily. The next time your Internet connection slows down, take your laptop to a place near the router that has a direct line of sight to the between the laptop and the router's antenna. If it's consistently up to speed but slowing down elsewhere in the apartment, then your problem is with your router's location in your apartment relative to where you do your computing. There are lots of ways to deal with this, like moving your router or getting a repeater/signal booster
Maybe something as easy as moving your furniture will do it. Really simple things like the positioning of your furniture can cause a wireless connection to slow down.
Another possibility is interference. Some wireless phones use the same frequency as wireless routers. Try a different channel (you can set this in your router configuration).
Another likely cause is your building's system -- your new apartment's Ethernet circuitry may be shared with a different set of neighbors than it had been previously, and these people may be much heavier users -- higher use means lower throughput. Older copper circuits only have so much capacity (GOD I love FiOS! -- Actually higher throughput than their advertised speed, and totally consistent!). Actually, from your description of the symptoms, this is a highly likely cause of your problems -- you may now be sharing a connection with some people who like to stream videos -- a big bandwidth hog.
You can test by transferring large files between two wireless computers sharing your router. Time it when your Internet is sailing along and time it again when it's getting bogged down. If the transfer times are the same, the problem is on the other side -- the building side -- of your router. There may not be much you can do about that except complain to the building management, and you'll need to weigh the pros and cons of taking that course.
Sometimes routers fail, but usually the symptoms are more catastrophic, like no connection at all. Sometimes it's just the wires -- have you tried getting a new CAT-6 cable from the wall to your router's WAN port? It's even possible that some electrical device is subjecting that wire to interference.
Anyway, good luck. I know this can be frustrating.