A “bum rap” means an undeserved criminal punishment. “Bum” means something of miserable quality; “rap” means a criminal accusation.
 
“Then another fellow told of how poor old Paddy Lyon got sent away on a bum rap” was printed in the New York (NY) Telegraph in 1907. “Bum rap” soon meant any false accusation.
 
“Bum steer” is a related term.
 
         
Wiktionary: bum rap
Noun
bum rap (plural bum raps)
1. (slang) A false accusation, or an injustice, especially one that leads to imprisonment.
2. (idiomatic) An undeservedly unfavorable portrayal or reputation.
 
(Oxford English Dictionary)
bum, adj.
Of poor or inferior quality; bad, unsatisfactory, second-rate.
1878   Puck (N.Y.) 4 Aug. 10/3   We don’t run no bum actors, no hamfatters..dat Irving is jest a snide.
1881   Public Press (New Albany, Indiana) 11 Apr.  The bum politician is abroad, the dread harbinger of the approaching spring elections.
   
(Oxford English Dictionary)
rap, n. (and int.)
U.S. Criminals’ slang. A prison sentence.
1870   N.Y. Clipper 23 Apr. 18/3   Charles, son of Victor Hugo, has been sentenced to six months imprisonment and fined $600, for an editorial article in the Rappel. That’s a pretty hard rap.
(...)
slang (orig. and chiefly U.S.). A criminal accusation; a charge. Frequently with modifying word (see also bum rap n. 1). Also: an identification of a suspect prior to a criminal charge (now rare).
1903   H. Hapgood Autobiogr. Thief xii. 265   ‘What makes you look so glum?’.. ‘Turned out of police court this morning.’ ‘What was the rap, Mike?’ ‘I’m looking too respectable. They asked me where I got the clothes.’
 
(Oxford English Dictionary)
bum rap, n.
Etymology: bum adj. + rap n.2
slang (orig. and chiefly U.S.).
1. A false charge or conviction; an undeserved criminal punishment.
1913   Chicago Tribune 5 June 5/4   They’ll frame a ‘bum rap’ on me before I’ve been out of stir a week.
2. An unjust criticism; an instance of being undeservedly blamed for something.
1921   Chicago Tribune 30 Sept. 10   Poor Tommy O’Connor got a bum rap for the time being.
           
20 April 1907, The Enquirer (Cincinnati, OH), pg. 12, col. 6:
Experiences of Real Thieves.
NIGHT PROWLER DISCUSSES HIS PROFESSION AND TELLS OF HIS MANY INTERESTING ENCOUNTERS.
[New York Telegraph.}
(...)
Then another fellow told of how poor old Paddy Lyon got sent away on a bum rap.
(...)
It’s pretty tough to be settled for a five-spot when you arr guilty, so you can imagine how it is to be in there doing five-on a bum rap.
 
24 January 1913, Rockford (IL) Register-Gazette, pg. 4, col. 3:
Mr. Rapp, in the language of the light-fingered gentry, seems to have got another “bum rap” in the house at Springfield Thursday.
 
Google Books
July 1914, The American Magazine, “Boston Blackie Stories” by No. 6606, pg. 12, col. 3:
I knew the coppers were working a ‘bum rap,’ for Mac had been with me all night.
     
Newspapers.com
17 September 1916, Dayton (OH) Sunday News, “Some of the Terms Used by Convicts,” pg. 5, col. 5:
If the charge is false, it’s a “bum rap.”
 
Google Books
“Aw Hell”
By Clarke Venable
New York, NY: The Reilly & Lee co.
1927
Pg. 20:
“That’s a bum rap old sawbones give you, buck. I’m sorry, but I’ve got my orders.”
 
OCLC WorldCat record
Television Programming: The “Boob Tube” Takes a Bum Rap
Author: John E Jr Baird
Edition/Format: Article Article : No language available
Publication: Intellect, 104, 2375, 590-93, May/Jun 76
Summary:
Attempts to test objectively one of the most prominent criticisms—that television programming is imitative and restrictive of viewer choice—in an effort to determine whether that criticism is legitimate. (Author/RK)
 
OCLC WorldCat record
Bum rap on America’s cities : the real causes of urban decay
Author: Richard S Morris
Publisher: Englewood Cliffs, N.J. : Prentice-Hall, 1980.
Edition/Format:   Print book : English
 
OCLC WorldCat record
Trade policy with heterogeneous traders : do quotas get a bum rap?
Author: Kala Krishna; Ling Hui Tan; National Bureau of Economic Research.
Publisher: Cambridge, Mass. : National Bureau of Economic Research, 2007.
Series: Working paper series (National Bureau of Economic Research), no. 13040.
Edition/Format:   eBook : Document : English
 
Urban Dictionary
bum rap
unfair blame, unjust sentence (english slang)
It was a bum rap. Eddie didn’t steal those paintings.
by VAKI5 May 09, 2005