Found my problem. I was using a test account which contained a LOT of old test orders. The easiest reorder button available was on a logged order which was actually "cancelled" status, not "confirmed" or "created". Apparently cancelled orders can't be reordered, which is fine. In that circumstance, you might consider hiding or disabling the button sometime in the future.
I had three old orders which triggered the "Limit reached" message for no apparent reason (no limits are set upon any of the products or components), but once I manually re-assembled the order contents in a cart, that message stopped appearing. I'll take that as a good thing - if it re-appears, I'll tackle the issue then.
Thanks!