Phony colleges were definitely a problem (though not the only problem). Thelin's history of American higher education has some good reading on this point. He refers to diploma mills as a "distinctively American contribution" to higher ed.

More generally -- most of my books that touch on this are on loan, but here are the broad outlines -- The regional accrediting bodies that accredit most US colleges and universities took shape in the 1890s, as part of a general push toward standardization at all levels of education. Before this period, quality control at the tertiary level was basically nonexistent (and quality control at the secondary level mostly consisted of individual universities' systems for accrediting individual high schools, but that's another story). Anybody could hang out a shingle and call themselves a college, and it would have been difficult for the typical prospective student to tell a legitimate (but new or little-known) institution from a straight-up scam. The accreditation system initially served to provide a sort of imprimatur of quality that students and parents could trust.

ETA: However, that wasn't the actual issue that drove the formation of the accreditation system. Rather, it was the need of the universities to be able to tell which transfer credits to honor and which not to. Relevant excerpt from Harper's University: The Beginnings, by Richard J. Storr (University of Chicago Press, 1966) (focusing on the University of Chicago and the North Central Association circa 1900):

The North Central Association did not long confine itself to consultation. Soon after its founding, the Association established a commission on accreditation of schools, of which Harry P. Judson, a University of Chicago dean, was chairman and George N. Carman, sometime head of Morgan Park Academy, was secretary. It was Carman's question -- shall we accredit the colleges? -- that opened debate on a policy which put the Association in the forefront of the effort to maintain standards in undergraduate education. One of Carman's reasons for advocating college accreditation was the impossibility of drawing a line of separation between the secondary school and the college. About half of the combined school and college requirements for the Bachelor's degree, said Carman, might be and as a matter of fact were pursued in either school or college. Carman added that failure to co-operate in fixing and maintaining standards had led too often to an unseemly scramble for students, which sacrificed what was best for them to the seeming advantage of the college.