  We conduct a search for z>~7 dropout galaxies behind 11 massive lensing
clusters using 21 arcmin**2 of deep HST NICMOS, ACS, and WFPC2 image data. In
total, over this entire area, we find only one robust z~7 z-dropout candidate
(previously reported around Abell 1689). Four less robust z-dropout and
J-dropout candidates are also found. The nature of the four weaker candidates
could not be precisely determined due to the limited depth of the available
optical data, but detailed simulations suggest that all four could be
low-redshift interlopers. We compare these numbers with what we might expect
using the z~7 UV luminosity function (LF) determined from field searches. We
predict 2.7 z~7 z-dropouts and 0.3 z~9 J-dropouts over our cluster search area,
in reasonable agreement with our observational results, given the small
numbers. The number of z>~7 candidates we find in the present search are much
lower than has been reported in several previous studies of the prevalence of
z>~7 galaxies behind lensing clusters. To understand these differences, we
examined z>~7 candidates in other studies and conclude that only a small
fraction are likely to be z>~7 galaxies. Our findings support models that show
that gravitational lensing from clusters is of the most value for detecting
galaxies at magnitudes brighter than L* (H<~ 27) where the LF is expected to be
very steep. Use of these clusters to constrain the faint-end slope or determine
the full LF is likely of less value due to the shallower effective slope
measured for the LF at fainter magnitudes, as well as significant uncertainties
introduced from modelling both the gravitational lensing and incompleteness.
