One of my clients who has been using online banking with QuickBooks recently came across the error OLSU-1016 when trying to download new transactions. They had spoke with Intuit who did some troubleshooting by creating a new QuickBooks file and setting up one account for online banking. That file was able to download the transactions, so the next set was to remove all online banking information and re-setup the accounts for online access. This did not fix the issue, so they decided that the working file needed to be sent to Intuit to be fixed as there was data damage that could not be fixed over the phone.
I ran some test of my own on the file, by doing a verify, which did not detect any errors. I did notice however, that they also had credit card accounts that were setup for online banking. These credit card accounts were also from the same bank. I had them go through and add/match the downloaded transactions for those accounts, then deactivated the online banking services for their credit card accounts as well. Once all accounts were deactivated, we proceeded to add the online banking service back to the accounts and was able to download transactions again.
The support article that Intuit has at http://support.quickbooks.intuit.com/support/Articles/HOW16085 does have that listed as one of the steps, however it is not just bank accounts that need to be deactivated, but all accounts.