Where does the idiom "I got a ‘bum rap’,"come from?

The AHD gives the definition of "bum" as a back formation of "bummer", a tramp or vagrant. By extension, bad or undesirable. "Rap" has been well-established since the early 20th Century as meaning "talk". So bum rap would simply be bad talk.
Another meaning of "rap" is "punishment". So the term "bum rap" also means bad punishment.

Personally, I think there’s some leakage between "rap" and "rep" (as in reptutation, or what is known on the street as "creds"). So a bum rap would means bad talk AND an undesirable reputation OR a bad deal in the form of undeserved (as percieved by the ‘rapper’) punishment.

- PW
who like most Brits has a whole nother definition of "bum".

- PW

Rap can also mean "blame" or "criticize".

I think in the context it means something akin to unfair/undeserved criticism.

-Tim

something akin to unfair/undeserved criticism

I just heard on the news that some Senator (R) thought that the Administration was getting a "bum rap" in the media because of the Weapons of Mass Distraction debate. So I guess you’re right, Tim.

- PW