Stability criteria of partly uncertain linear systems with variable coefficients play an important part in various fields of robust control. In this paper we consider linear systems with periodic coefficients that may vary within their known lower and upper bounds depending upon time as well. Stability analysis of systems with variable coefficients is often based on the Lyapunov functions method leading in many cases to rigid sufficient stability conditions. A new approach to stability analysis of such systems is described in this paper. Using this method we obtain an upper bound for the real parts of the characteristic exponents of the system solutions yielding sufficient stability criteria and find the cases in which these bounds are reached. This explicitly determines the "worst" possible periodic coefficients yielding the largest value of the maximal real part of the characteristic exponents. Certain results are extended to the case of nonperiodic time-dependent bounded coefficients.