We study the order of accuracy in time discretization for poromechanics, applying the fixed-stress split scheme. High-order methods in time are often used, for example, in order to match high-order schemes in space. We consider two operator splitting methods motivated from the fixed-stress split: a two-pass algorithm and a spectral deferred correction (SDC) method. The two-pass algorithm, known for a symmetric operator splitting scheme as Strang's splitting, is typically employed for a high-order method of non-stiff ordinary differential equations (ODEs). The SDC method is also a structure conserving efficient scheme for an arbitrary high-order of accuracy. However, poromechanics yields the governing equations that have a form of differential algebraic equations (DAEs), where typical high-order time integration schemes for non-stiff ODEs are not appropriate. Performing mathematical analysis, we find that both methods do not achieve high-order accuracy in poromechanics, although both methods can reduce the errors in their time discretizations. From numerical experiments, we find that the mathematic estimates are consistent with the numerical results of the fixed-stress type two-pass and SDC methods, which still provide the first-order accuracy in time.