For how Windows determines internet connectivity, see this SuperUser question. Basically, Windows tries to access a Microsoft hosted text document and judges your connectivity depending on the response (timeout, 403 denied, 200 OK, etc).
Yes, BitTorrent (or anything else) using excessively large chunks of your bandwidth can saturate your connection. Basically, you're transferring so much at once that other packets end up timing out and getting dropped. Imagine a water pipe, if you try to put too much in it will fill up faster than it can be emptied and start overflowing, losing the water that overflows.
A typical IP (Internet Protocol) request requires you to send some packets to the server you are requesting from, which will then send some packets back - its response. This requires both some upstream and downstream bandwidth. When your upstream bandwidth becomes saturated, you can't send the request. When your downstream bandwidth becomes saturated, you can't receive the response. It is entirely possible for a BitTorrent client to use all your available bandwidth.
Generally, it is best to set your BitTorrent client to only utilise up to 80% of your upstream and downstream bandwidth, as determined by speed tests. If you require low latency, e.g. online gaming, the percentage should be even lower.
It's also possible for BitTorrent to overload home modems and routers by opening too many connections, overflowing their NAT tables. The maximum number of connections should therefore be kept at a fairly low level. A maximum of 300 connections globally should be alright. An overflowing NAT table's symptoms will vary from router to router, but often cause them to freeze.
Best Answer
The distinction here is bits per second and bytes per second.
Your internet connection speed is 256 kbps (note the small 'b'), which is 256 kilo bits per second.
There are 8 bits in a byte, which is how most applications will report how they're downloading. So 256 kbps is equivalent to 256 / 8 kBps which is kilobytes per second which is 32 kBps. In this case, note the large B to denote BYTES.
Your internet connection of 256 kbps is raw bandwidth. Every time you make a connection, there is supplemental information sent and received. These are generally protocol specific headers / additional information (e.g. TCP headers / HTTP headers) and they add overhead to the overall communication reducing (normally only by a small amount, but still reducing) the total amount of bandwidth you have available for raw downloading of data.
http://en.wikipedia.org/wiki/Bit_rate
http://en.wikipedia.org/wiki/Bandwidth_(computing)