No, they do not wear out, unless you are constantly plugging/unpluggin cables into it, or turning it on and off all the time. Electronic devices rarely wear out, just the mechanical parts (I’ve worked as a repair tech in the past, and also have had lots of devices). I’m still using my original Linksys wired router from over 5 years ago, and an old Linksys wireless-B router for several years.
Now, they do have problems sometimes, and if you don’t know about esoteric TCP/IP tricks (such as IP address lease times, releasing and renewing the address, etc), it may take a while to figure out. Netgear seems to be particularly prone to problems connecting in the correct sequence. You really need to disconnect it, reset the modem, wait for the modem to finish booting up and stabilizing, then connect the router to the modem and turn it on, let it stabilize (much quicker than the modem), then connect the PC and turn it on (this is all from memory, but I think that’s the drill).
Sometimes it may not work until the next day unless you know how to use the command-line and “ipconfig”.
I think these sort of problems are what make people think it’s “worn out”. Same with computers – they get so much junk on them that people buy a new one, not knowing that they can easily restore the computer to its original like-new condition with the recovery disk.