In what cases does it make sense to buy a plane before you begin pilot training?
I will mention some considerations that were not mentioned above. First of all, it does make sense to buy a plane for training, and it can save you a lot of money on rent but you need to be aware of what you are doing. The positive side is that airplanes do not depreciate as fast as most assets of their kind (it is not unheard of being able to sell an airplane for practically the same amount as you bought it several years ago). However, consider that:


Usually an airplane you want to train in (something sturdy and easy to control, like a C-150) is not an airplane you would like to fly most of the time (traveling in a 150 is not a lot of fun, now a twin Comanche is a different story...).
Insurance costs for a student pilot (even with various restrictions) can nearly negate the advantage of not having to rent.


So, if you would like to buy an airplane to mostly train (and have some fun occasionally) and then sell and buy a 'travel' (or aerobatic) airplane, it can save you money. It is better to get your private first, however, so you can concentrate on initial training (which is a handful) without worrying about the 'pleasures' of plane ownership.