I mentioned suspicious signals from the encoders. Here's what I was seeing. This is a plot of the distance traveled versus time. The plot is showing that one of the wheels is able to travel more than 300m in a fraction of a second. Cool.
The cause of the problem was a bad solder joint on the encoder's ground wire. Intermittent contact was causing flickering of the signal and the mbed is easily fast enough to count them all.
A quick hit on all the encoder solder joints after liberal application from the flux pen fixed it up real nice.
It now appears my GPS is toast. Seriously, universe?!