If the model is not significant when two predictors are entered together, could it become significant if they are entered separately? These predictors are highly correlated.
In another (significant) model, when the predictors are ran separately, one of them accounted for higher unique variance (around 40%) than the variance accounted for when they were entered together (around 10%). I am also wondering why that is?
 A: TLDR: This question is inspired by the $R^2$ numbers reported in an article about the effects of socioeconomic status (SES) on brain development in children. It turns out that some $R^2$s are incorrectly reported: the typos are not in the main results table and don't change any of the conclusions but there is an apparent inconsistency which the OP noticed.
I contacted the article's first author, Prof. Noble who is at Columbia University, and she was kind enough to look at the data records and reply. She confirmed there is a typo in the $R^2$ change numbers reported.

Instead of 0.59, the $R^2$ change should have been reported as 0.059. The Beta and p-value are correct. So the sentence should read as follows:
"In the amygdala, when controlling for age, total cortical volume, and gender, parent education alone accounted for significant unique variance, with lower education levels associated with larger amygdala size (R2 change = 0.059, Beta = 0.286, p < .002; see Figure 2)."


As @Dave explains (+1), it's theoretically possible to add noisy variables which nevertheless use up degrees of freedom, so that the bigger model is "worse" than the smaller model. This is called overfitting.
I'm more interested in the second part of the OP's question. I paraphrase:

In a model Y ~ covariates + X1, the predictor X1 accounts for 40% of the variance in Y. But in the model Y ~ covariates + X1 + X2, the two predictors X1 and X2 together account for only 10% of the variance in Y.

The OP explains in a comment that these numbers come from the following study of the effects of socioeconomic status (SES) on brain development in children:
Noble, K.G., Houston, S.M., Kan, E. and Sowell, E.R. (2012), Neural correlates of socioeconomic status in the developing human brain. Developmental Science, 15: 516-527. https://doi.org/10.1111/j.1467-7687.2012.01147.x
I suspect that the numbers are mistakenly reported by Noble et al. Or perhaps I don't follow their calculations. In any case, here are the numbers which don't seem to make sense. I combine data from Table 2 and the text below Table 2 on page 6.




brain region (Y)
regression
R-squared




amygdala
Y ~ age + volume + gender
0.634



Y ~ age + volume + gender + income
0.634 + 0.001



Y ~ age + volume + gender + education
0.634 + 0.59



Y ~ age + volume + gender + education + income
0.634 + 0.088


hippocampus
Y ~ age + volume + gender
0.457



Y ~ age + volume + gender + income
0.457 + 0.44



Y ~ age + volume + gender + education
0.457 + 0.02



Y ~ age + volume + gender + education + income
0.457 + 0.124




Note that one R-squared ends up greater than one:  0.634 + 0.59 = 1.224. This is impossible.
So I suggest that both the 0.59 and the 0.44 are mis-reported. It seems likely that the correct $R^2$ change from the baseline step-1 regression are 0.059 and 0.044, for the amygdala and the hippocampus respectively. Those numbers would be consistent with the full models Y ~ covariates + eduction + income because the $R^2$ changes from adding each predictor separately add up to at most the combined $R^2$ change, as expected: 0.001 + 0.059 ≤ 0.088 and 0.044 + 0.02 ≤ 0.124.
A: If including the second variable improves the fit only a tiny amount, yet the degrees of freedom change, that could happen.
I find it easier to take this to the extreme. Add in one variable that really matters and a bunch that do not, say 100. Those 100 irrelevant variables will contribute to the degrees of freedom in the $F$ or $\chi^2$ distribution, making quite high the threshold for model improvement to be significant. However, those 100 irrelevant variables make little difference to the model fit, not enough to make up for the degrees of freedom. Thus, those $100$ irrelevant variables wash out the real effect of the one relevant variable.
For just two variables, the same idea applies, just less dramatically.
