Bath is probably the second or third most popular tourist city in England. (Only London, and possibly York, have more visitors). So there are certainly plenty of B&Bs. The problem is that there are also plenty of visitors wanting to stay in them, which pushes prices up. There aren't that many places within your price bracket, so it's possible that they might all be full if you don't pre-book.
To get an idea of what's available, browse through the accommodation listed here:
http://visitbath.co.uk/site/where-to-stay
Chris