I think I finally figured out why everyone keeps having this problem. 

I have hundreds of clients and receive hundreds of checks, some of which will process correctly using the check payment process in Quickbooks and others that give the dreaded "The transactions was declined because the bank account is incorrect, or there is a security concern." I had enough of these error checks together at one point that I realized something peculiar about them while troubleshooting. There were checks from different clients, that would seemingly have no reason to know one another, that shared identical account and routing numbers on their bill pay checks. I believe that the banks are using a central routing and account number as a cryptographic measure by which these routing and account numbers are a "public key" while the bank possesses the "private key." In other words, the account and routing numbers you have on the bill pay checks are not individually tied to the client, but are tied to an account to represent a number of the bank's clients, and then when the check is scanned or deposited with these numbers, they know which accounts to draw from in the bank's system. I think the bank is doing this to protect the real routing and account numbers of their clients from potentially falling into the wrong hands. I also think that Quickbooks' system is realizing this to an extent or at least able to know that there is not one name attached to these accounts and is declining it based on a security concern or the numbers being inaccurate (because they don't tie individually to the customer but a multitude of customers). 

For example, I had 4 checks, each in identical physical appearance, with 4 different customers that would have no reason to know each other let alone share financials, and only had 2 sets of identical routing and account numbers between them. That's simply not a coincidence.