HAve you setup your Bittorrent client to limit the connection speed it is using ?? -- If you leave the upload setting to unlimited it will saturate the available bandwidth causing you to not be able to do other things and also slowing your download speed -- you have to remember that to use your connection there has to be some available upload speed left even for downloading because there has to be some communication between your system and the other systems and if your bittorrent is using all of the available upload bandwidth there is no overhead to keep the connections from dropping. Adjust the settings to keep at least 20 - 35% of your bandwidth available for the overhead (ie. if you get 1.5Mb\s upload capacity limit the bittorent client to around 128KB\s +\- (approx 1Mb\s) - by doing this you keep part of your upload bandwidth available for the download negotiations and for other things like Browsing the internet while it is running. (after doing this you'll also probably get faster downloads than you would with it set to unlimited upload speed)
If the upload bandwidth is saturated when you try to browse the internet or do anything else it will give you a timeout error because it is unable to send the requests to the server you are trying to connect with. When you attempt to browse are you getting timeout errors ?