Depending on where you are, but: OVH is known to offer dedicated servers with very generous bandwidth allotments for very little money. E.g. their Dutch budget brand http://www.isgenoeg.nl offers a basic server with 5TB, then 10Mbps bandwidth for EUR 15/month, and their UK http://www.kimsufi.co.uk/ is similar.

(No personal experience, offer seems valid for nationals only, WebHostingTalk suggests there's little support, OVH apparently change their upstream providers all the time to get the best possible deals - you get a lot of bandwidth but latency varies from day to day. Seems acceptable for a simple file server.)