I agree that a too small of a bike can cause back problems, having a bike that fits properly--as you already know--is very important.

Also take in consideration that mountain biking uses your muscles differently than road riding. Whenever I take newbies out for trail rides they often complain about back and shoulder pain--usually from using those muscles in a new way. Part of that is their core muscles aren't conditioned to that style of riding. It's typical for new riders to use the death grip on the handlebars—nerves causing them to tense up. That tension causes stress on the muscles (especially shoulders and back).

Another cause of back pain is a bad pedal stroke--pulling too much on the upswing. It's common for a lot of us that use clipless pedals to adapt that type of pedal stroke. How many times have I been told to pull up on the the pedal! when climbing? A million times. But that can actually be really bad for your back. Here is a good blog post on pedal stroke.

One more common cause for back pain: tight hips and leg muscles. Here are some stretching tips.

Welcome to the world of trail riding!!!