weighted mixed model regression

Dear all,

I have a question about lmer's verbose output. Can one derive from this 
verbose output that the estimates that lmer has produced are 'reliable'? 
I noticed that lmer yields estimates for the fixed and (correlated) 
random effects, where other programs (spss, mlwin) fail to converge. 
However, often in such instances, the smallest eigenvalue of the 
estimated (co)variances of the random effects (extracted using the 
VarCorr function) was very very close to zero, something like 1E-15, 
meaning that this matrix is close to singular. Also, one of the elements 
in the verbose output was very close to zero. Does this imply that the 
verbose output is lmer's way of saying "there may be something wrong 
with the estimated covariances of the random effects"?

I ran a lot of, say 'problematic', models lately, first in lmer and then 
in spss. Mostly, if lmer has a close to zero figure in the verbose 
output, then spss doesn't converge and speaks of 'redundant elements' in 
the set of random effects. But also, there was a model in which spss 
didn't converge and showed this redundancy message while lmer DID 
converge and did NOT have an extremely small (close to zero) value in 
the verbose output. My conclusion was therefore that, for that 
particular model, the estimates of lmer were 'ok'. Still, however, in 
general I'm in doubt  about how to deal with the information in lmer's 
verbose output. Could anyone help me out, please?

Ben.