Why Being Denied Admissions Over Accreditation is Worse Than Just Being Denied
Grad school acceptance or denial is a tricky thing. You never know what they consider and don't, or what the weight of those considerations will be. You don't know what your competition is, and you have no idea how much of the competition they plan on taking. You get little help from the department and then get little feed back. In short it would be a very tricky thing to assess how many graduates from Bob Jones University get denied grad school admission based upon accreditation. Normally you just get a nice letter saying better luck next time.
For grad school any number of things can get in the way of acceptance. You could be unqualified for that school. Maybe your GPA is too low, or you didn't do well on the GRE or perhaps you got your undergrad in a field too far away from the current one you are applying for. This isn't the end of the world. Take a few classes from the community college or even at that college and you could turn your deficit around in no time.
You could also be qualified, but there just wasn't enough room. Some programs take only a small handful of applicants. A rejection letter in this way is more of "try again next year." It isn't that they didn't like you, they just didn't like you enough to offer you the spot.
Neither one of those is easy to take, but they are manageable and you can turn either one around with a bit of luck and perhaps some volunteer work or studying. But being rejected because of accreditation hurts a bit worse then either of the above. A letter saying that they don't recognize your undergraduate degree is like saying, "Please don't ever apply here again." They've totally rejected all the work you did. It isn't that you were out competed or simply in the wrong field. No you were in the wrong ball park. You can't get up to bat if you are standing in the bleachers watching with dashed hopes.
Further, being rejected because of accreditation makes you apprehensive of applying just about anywhere. You constantly have that nagging feeling in the back of your head, "No matter how good I look, no matter how qualified I can be, I could still be rejected sorely because of my undergraduate degree." Frankly, I'd take flat out rejection then rejection because of accreditation any day. At least there is some recourse through rejection. Raising your GPA, taking more classes, volunteering, retaking the GRE, or applying the next year I can do all of that, but I can't undo my undergrad degree.