There is a significant number of planes flying with bad aircraft addresses
There are indeed, although a "bad address" won't necessarily duplicate one belonging to another aircraft (there are 16 million possible combinations).
But there are certainly documented cases of aircraft airborne simultaneously in the same airspace (including the London TMA) and sending the same address. While that would affect the Mode S information that the controller sees, both aircraft would still be responding independently to Mode A and C, so radars that use code-to-callsign conversion should still be able to identify both.
Incidentally, one other possible effect of Mode S duplication is to confuse TCAS, which uses the 24-bit address under certain circumstances to determine what type of RA is issued (for example two aircraft at the same altitude).