Properties of the best achievable performance of sampled-data control systems are investigated in this note. Contrary to our intuition, the best sampled-data control performance does not always converge to the best continuous-time control performance, even as the sampling period approaches zero. A counterexample is presented and a lifting-based analysis is performed, It is shown that the performance convergence is guaranteed if, and only if, a certain Hankel norm and an aliasing effect computed from a specified antialiasing filter tend to vanish with a decreasing sampling period. It is also shown that this condition is reduced to a simpler form if the specified filter has special forms, These results can be generalized to a system involving a generalized hold.