Hi B Miner,
Assume that Y = X1b1 + X2b2 + e where the independent variables are i.i.d and the residuals e is orthogonal to X=(X1,X2). Assume that you do not have access to X2. Hence, let u = X2b2 + e. Finally, assume that b1 = 0.
Estimate the following equation : Y = X1b1 + u. There will be a relationship between u_hat and Y because u is orthogonal to X1 but no relationship between u_hat and Y_hat = X1b1_hat.
More generally, I believe that this will be the case whenever you plot the residuals against some orthogonal variable - i.e. orthogonal to the variables that generate variation in your Y's.
Am I right ?