Despite the proof of identity requirements, remember exchanges and wallets don’t provide the same protections banks do.
For example, there is often no or limited insurance for your account if the exchange goes out of business or is robbed by hackers, such as was the case with the infamous failed exchange Mt Gox.
Bitcoin does not have legal status as a currency in most of the world, and authorities usually do not know how best to approach thefts. Some larger exchanges have replaced customer funds after a theft from the exchange itself, but at this stage, they are not legally obliged to do so.
