This is refer to as multicollinearity, that is,  when the predictor variables linearly related which should not be, it can be detected using VIF (Variance Inflation Factor). You can read more about multicollinearity.

The dependent variables are very likely to have some correlation in cross-sectional regression, unless a design for the research specifically eliminates cross-sectional correlation.  This multicorrelation can be eliminated in some circumstances by a process called "blocking".  The technique is used frequently in agricultural experiments.  When correlation does not exist in the dependent variable matrix that matrix is orthogonal.  If the variance of the estimates do not change with the magnitude of any of the dependent variables the diagonal of the regression matrix is a series of ones *1s" along the diagonal of the matrix and all other cells are zero.  This results in unbiased estimates of the regression coefficients and also their standard error.  The regression meets the "Gauss-Markov" conditions for an unbiased estimator.  However, the distribution of the estimates of the coefficients may not be distributed normally.  If an important dependent variable is left out of the regression and is correlated with one of the dependent variables, the estimate of that coefficient will be biased.

Many thanks Ajibode and L. Brown for response