Barit1 has it - look up his post.
The issue is that if you have air at 10-20 atmospheres then the partial pressure of oxygen will be equivalent of 2-4 atmospheres. Things tend to burn very well in high pressure oxygen, and will start to ignite at a lower temperature. Plainly a tyre filled with oxygen is the equivalent of a small bomb. The question then being, what causes it to go off?
I'll confess I'm not sure of the answer to that, but I guess you could get a static spark / spontaneous combustion of hot volatile compounds..?