What irks me about the NCAA requesting bank statements is that they most likely are going to try to make generalizations based on summary information. You really can't get much detail on deposits from any bank statement that I've seen. Sure, some of the deposits are going to have very high level information like (hopefully not in this case) "Mississippi State University Treasurer" or something like that in the event it was a wired deposit. Unless I'm mistaken, cash/checks deposited are just going to say something very generic like "deposit". So, if the Sidney's were receiving payments that would compromise eligibility (from agents, boosters, etc.), it's not going to show up barring anything completely retarded like a check being cut from a University's treasury or direct deposited from an agent's firm. Shady transactions are going to be dealt with in cash and if your only information is a bank statement - you aren't going to be able to trace much more that what's given to you there.
But, what the NCAA would probably do is try to establish how much of a balance the Sidneys should have in their accounts based on their documented legitimate income sources. Anything above that would be "assumed" to come from illegitimate sources, unless proven otherwise, from the NCAA's perspective and clearance would be denied. I have no idea what sort of authority the NCAA can or can't do regarding clearance of players, but I assume if they wanted to deny clearance on this alone, they could if they wanted to. The Sidney's would have no choice but to prove where the unexplained deposits came from if they wanted Renardo to play college ball. If there was anything illegitimate there, they wouldn't want to even try defend themselves as I'm sure none of the cash payments were reported to the IRS as income. That's another can of worms they don't want to open.
Even if the Sidneys are sqeaky clean, it just makes sense not to hand anything over. That's just more information available for the NCAA to misinterpret. But, seeing our luck with the NCAA, I'm sure they'll mark this up as a huge risk factor and will weigh heavily in their ultimate decision.