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In this paper we consider computing estimates of the norm of the error in the conjugate gradient (CG) algorithm. Formulas were given in a paper by Golub and Meurant (1997). Here, we first prove that these expressions are indeed upper and lower bounds for the A-norm of the error. Moreover, starting from these formulas, we investigate the computation of the l2-norm of the error. Finally, we define an adaptive algorithm where the approximations of the extreme eigenvalues that are needed to obtain upper bounds are computed when running CG leading to an improvement of the upper bounds for the norm of the error. Numerical experiments show the effectiveness of this algorithm.