First off, it is not clear that Emilia's specific problem is being caused
by over-parameterization. Emilia, could you perhaps give more information
about the nature of the dataset that you're analyzing? Is it a 2x2
within-subjects, within-sentence balanced design without a great deal of
missing data? In my experience with the last few pre-1.0 versions, lme4 is
generally very good at converging to an optimum for these kinds of datasets
with the number of observations and groups your fitted model reports. Have
you tried fitting the model with the nlminb optimizer, either by including
optimizer="optimx",optCtrl=list(method="nlminb")
in the list of arguments to lmerControl, or by using the last pre-1.0
version of lme4 (available as lme4.0 on R-Forge)? Do you still get similar
problems with the nlminb optimizer? (You should definitely not get the
result that the simpler model has a higher log-likelihood.)