I appreciate the replies I have received so far. One thing I failed to emphasize is the intermittency of this problem. It appears to be related to the Messaging system in a subtle way. If FindFit doesn't have a problem (as it apparently did not on some of your computers) it does not try to throw a message and the fit parameters get returned correctly. The particular example I pasted does not converge in 500 iterations (on my computer) and FindFit tries to throw the Message. I think it may be a failure of Message, not the function itself, that is the source of the problem.
This morning, I have a fresh session and I deleted the cache. When FindFit throws the message, a pink Message notebook opens with the message, as expected behavior. I think somehow I buggered the windowing system, so Mathematica gets confused as to where to put the Message. Maybe it is out there somewhere, I just do not see it.
This is very hard for me to diagnose. For now, I have reduced the Accuracy and Precision goals and keep my fingers crossed. However, since I am developing this as a tool for other engineers, I have to find a resolution for this.