Lets assume we have a 4 bit microprocessor (4 bit ALUs, 4 bit registers) and an address bus also 4 bits wide. Everywhere I read says that it can access 16 bits (2^4) of memory. Maybe I'm reading it incorrectly though isn't that completely incorrect?
This is the maximum, yes. More than this would involve such silliness as reading into more than one register. You really don't want the smallest addressable datum to be larger than the largest register.