Recently, the issue of robustness of adaptive filtering algorithms has been investigated using the H-infinity paradigm. In particular, in the constant parameter case, the celebrated (normalized) LMS algorithm has been shown to coincide with the central H-infinity-filter ensuring the minumum achievable disturbance attenuation level. In this paper, the problem is re-examined by taking into account the robust performance of three classical algorithms (normalized LMS, Kalman filter, central H-infinity-filter) with respect to both measurement noise and parameter drift. It turns out that normalized LMS does not guarantee any finite level of H-infinity-robustness. On the other hand, it is shown that striving for the minimum achievable attenuation level leads to a trivial nondynamic estimator with poor H-2-performance. This motivates the need for a design approach balancing H-2 and H-infinity performance criteria. In this regard, an illustrative example is presented showing that the central H-infinity filter is best suited to achieve a satisfactory tradeoff between the two performances.