If what you've given in the thread so far is accurate, you can already eliminate any issues from Blizzard or your ISP, you have another computer in the house that is working just fine.
This takes it down to just a few possibilities:
If both of the computers are wireless and in the same room:
Your computer may be have a bad wireless adapter, or there's interference between you and the router. There is also the possibility that windows is acting up and putting your wireless card to sleep after X amount of inactivity.
If both of the computers are wireless but in different rooms:
You may not have as good a connection on the PC that's having issues, or something is causing interference (microwaves, refrigerators, any high powered device can cause it) intermittently causing your connection to drop momentarily from the router.
If the good PC is wired and the bad PC is wireless:
You may be experiencing the same things described above, but you can also consider that the router's wireless antenna is on the fritz (happens a lot with cheaper routers). If your router came from your ISP, call em up and have them walk you through troubleshooting. If they find something wrong, they will usually replace it for free.
If the good PC is wireless and the bad PC is wired:
The ethernet port you're plugged in on the router may be going bad, Try another ethernet port on the router. The Ethernet cord itself may have a fractured wire or a bad end. Try another ethernet cord from the router to the PC and see if this helps. If neither of those helps, the router itself may be going bad.
If both of the computers are wired:
Swap the ethernet cables between the computers and see if the good PC has issues, and if the bad PC is fixed. If the bad PC is fixed, but the good PC is now having issues, try moving the ethernet cable it is connected with to another free slot on the router. If it's still having issues, replace the ethernet cord and see if it works.
If after swapping the cables between the two the issue still exists only on the bad PC, then you may be having an issue with your network card or windows configuration. A bad driver for your network card can sometimes cause this issue or windows could just be ignoring inactivity settings, or your computer's BIOS could automatically be cutting off the network card under certain circumstances. At this point, you might as well have a professional of some sort take a look at it.
Hope this helped