How to deal with correlated fixed effects in a mixed-effects model? I am trying to predict (binary) memory for pictures based on two continuous fixed effects: memorability and clutter. Using a mixed effects model, I found that both effects predict memory and including both in a model is better than either on their own.
However, memorability and clutter are highly correlated. 
1) How do I interpret my result (that including both is better), given that they are correlated? Can they still be independent predictors, yet be correlated?
2) Is this a problem for the model in general? How do I deal with correlated fixed effects?

In R:
m_mem <- lmer(memory ~ memorability + (1|subject), data=memDat, family='binomial')
m_clut <- lmer(memory ~ clutter + (1|subject), data=memDat, family='binomial')
m_mem_clut <- lmer(memory ~ memorability+clutter + (1|subject), data=memDat, family='binomial')

anova(m_mem_clut,m_mem) and anova(m_mem_clut,m_clut) are significant.
[Edit: R dput can be found here ]
 A: First of all, I would not consider a correlation coefficient of 0.383 to be too high. It is not troublesome at all, but only indicates moderate correlation. Given that both ANOVA tests reject the null hypothesis that the smaller model might suffice, it seems that both variables provide valuable information in explaining the response, even after taking the effect of the other predictor into account. Also, the AIC of the model with two predictors is smaller than the ones of the smaller models.
A few general remarks: Whether highly correlated regressors pose a problem depends on the aim of the investigation. If the goal is prediction, then one does not need to fear much: It can be shown that the standard errors of prediction for the response are not seriously affected by correlation among the predictors. However, correlation induces a strong upward shift in the standard errors of the regression coefficients. They become highly unreliable. Intuitively, it should be clear that one cannot correctly attribute the effect of a given regressor in presence of high correlation. 
