I may lose my own bet. By hand I computed less than 50 binary words of length 10 that start with 0. A little less than 40 of them have 4 or 5 of the subwords. This should be easily handled by computer, and with some patience can be completed by hand. My guess is that 21 is close to the maximum, and that there will be less than 10 words of maximal length.
Gerhard "Just Keep Adding Another Digit" Paseman, 2012.09.19