the simpler model does not converge and is singular scores <- c(26, 25, 14, 13, 10, 8, 22, 13, 10, 8, 8, 5, 
            30, 25, 10, 10, 7, 5, 43, 31, 12, 8, 2, 4,
            45, 31, 31, 10, 10, 4, 53, 23, 10, 1, 1, 1, 
            12, 16, 15, 16, 37, 28, 21, 20, 21, 15, 21, 17, 
            23, 27, 47, 27, 24, 59, 9, 23, 21, 12, 13, 14, 
            18, 17, 13, 18, 19, 21, 14, 14, 17, 21, 12, 13)
dat <- data.frame(Subject=rep(c(1,2,3,4,5,6), each=6), 
           Condition=rep(c("A","B"),each=6*6), Score=scores, 
           Day=rep(c(1,2,3,4,5,6),times=6))

I thought convergence and singularity issues mostly arise due to overly complex models. However it seems in my case, a simpler model is more problematic than a more complex one:
Simpler model (fails to converge, singular fit):
fm <- lmer(Score ~ Condition + (Condition|Subject), dat)
boundary (singular) fit: see ?isSingular
Warning message:
Model failed to converge with 1 negative eigenvalue: -9.0e-01 

More complicated model (no errors or warnings):
fm <- lmer(Score ~ Day*Condition + (Condition|Subject), dat)


Seems like the folk theorem of statistical computing might apply here, as it nearly always does.

When you have computational problems, often there’s a problem with your model.

Have you tried plotting the data?

It looks like both conditions differ pretty much, so your "more complicated" model in fact makes more sense.
You also don't have much data, which can lead to problems. The simple model to consider first might be the one without random effects.
