Abstract <p>Upper bounds for the Bures distance as well as for the trace-norm distance between quantum Gaussian states of CCR in terms of the mean vectors and covariance matrices are obtained, basing on a fidelity-like quantity which we call the states overlap. We argue that these bounds are better adapted to the Bures distance and hence to the corresponding methods of state estimation. We also briefly recall our results concerning bounds for the states overlap for fermionic Gaussian states of CAR. The problems studied in this paper can be considered as a noncommutative analog of estimation of the Hellinger distance and the total variance distance between Gaussian probability distributions in the classical probability theory.</p>