When does a p-value (standard error) of a linear model coefficient decrease with increasing levels of categorical predictor variable? Why this happens? I fail to see how collinearity and/or regressing residuals on one of the variables  is relevant for contrasts of categorical predictors.
Suppose we have:
dat <- data.frame( label = rep(LETTERS[1:3], each=4), value = c( 1, 0.96, 0.96, 1.03, # A 0.74, 0.45, 0.01, 0.89, # B 1.00, 1.02, 1.04, 1.06 # C ) )
Notice that one value is suspicious for level B. In any case, the overall mean of group B seems slightly lower, too. Now, then:
round(coef(summary(lm(value ~ label, dat[1:8, ]))), 3) # levels A and B # Estimate Std. Error t value Pr(>|t|) # (Intercept) 0.988 0.137 7.182 0.000 # labelB -0.465 0.194 -2.391 0.054 round(coef(summary(lm(value ~ label, dat))), 3) # levels A, B, and C # Estimate Std. Error t value Pr(>|t|) # (Intercept) 0.988 0.113 8.777 0.000 # labelB -0.465 0.159 -2.922 0.017 # labelC 0.042 0.159 0.267 0.795
The p-value for level B has decreased 3x for the second case. If anything, I would expect it to increase, e.g. to counteract the inflation of family-wise error rate due to multiple testing.
I can't wrap my head around that if A were my baseline condition (control), without going bayesian, I could indefinitely increase my confidence in treatment B to have an effect by just testing more treatments (C, D, E, ..., Z) without ever increasing the sample size in group A nor B.