Hello. I am experiencing two problems that make Internet access quite difficult.

First, my computer has recently, for no apparent reason, stopped detecting the connection of an ethernet cable. If I navigate to System Settings > Network > Wired, the status reads "Cable unplugged." While it is possible that the port sustained damage, this problem was first encountered while the computer was in use as a sudden interruption of connectivity.

Second, my wireless connection terminates every few minutes. To restart it, I must execute the following commands:
modprobe -r iwlagn ; modprobe iwlagn ; iwconfig
I run these commands with a period of 1 minute (i.e., one execution per minute) as a root cron job. While this action definitely prolongs the life of the wireless connection, it still fails about once per 1-2 hours.

I recently tried upgrading my kernel to 3.10 using the debian backports repositories, but the problems were not resolved. Also, I have not found the output of dmesg to be particularly instructive, but I could provide it via a pastebin if it might be helpful. Thanks very much for any advice you could provide.

miscellaneous system details that might be relevant to a solution:
Debian GNU/Linux 7.1 (wheezy)
Linux 3.2.0-4-amd64 #1 SMP Debian 3.2.46-1 x86_64 GNU/Linux
02:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168B PCI Express Gigabit Ethernet controller (rev 06)
08:00.0 Network controller: Intel Corporation Centrino Wireless-N 1000 [Condor Peak]

Maybe this is something as simple as a dodgy cable? have you tried replacing it?

Thanks for your suggestion. I did try several other cables and ethernet jacks, which are functional on my coworkers' machines, so I think that the problem originates either with my hardware or firmware.

Well worth a try!
I would tend to agree with you if you had not installed any new software when the problem started it is probably a hardware problem - Time to hit the credit card!