Take the following example:
I wish to predict physical performance as a function of height and weight. I already know weight negatively affects performance. Height also negatively affects performance, but I don't know if this is only because of the weight, or if there is a positive correlation between height and performance when disregarding the correlated weight. So, I measure the heights and weights of thousands of individuals, and measure their performance. I then model performance with multivariate regression: $p = (a * \text{height}) + (b * \text{weight}$). The estimated coefficients $\hat{a}$ and $\hat{b}$ turn out to be 0.2 and -1 respectively when using "lm" in R. The individual significances of these estimated coefficients are very high according to the t-tests (as is shown in a table when using the R command "summary" of a linear model made with the command "lm"), and their std. errors are very low. I therefore conclude that height is positively correlated with performance when disregarding the negative effects of weight. When is this conclusion a mistake?
To rephrase, if I have plenty of data and the std. error is very low for the estimated coefficients of both correlated variables (say $x_1$ and $x_2$) in the regression summary produced by the R command "summary", is there any reason (in general) why I should distrust the estimated coefficients for $x_1$ and $x_2$? I.e. can the collinearity mean that these estimated coefficients do not necessarily reflect the real coefficients even though the standard errors shown by the regression summary are very low?