A concept of the vector space of imperceptible observation errors in linear Gauss-Markov models with uncorrelated observations, initially proposed in the earlier work of the author, is presented together with some improvements and new developments. The gross errors falling into that vector space pass absolutely undetected through all possible statistical tests set in the least squares estimation and unnoticeably distort the resulting values of one or more of the model parameters. The relationship is established between the concept of imperceptible gross errors and the concept, proposed by other authors, of the gross errors which can be detected but not identified due to specific properties of a network's structure. The theory is illustrated with a simple numerical example.