People don't "usually" combine active and passive filters in the way you're suggesting, but it's certainly possible. It's not generally a good idea, but it's certainly possible.
Fairly common is a stage of passive filtering followed by high-order active filters. In this case, the first stage filters out-of-band noise without worrying about the active element being driven non-linear. The relatively clean output is then handled by the active elements.
Except in unusual cases, a 5-stage LC filter can usually be done better by an active equivalent, particularly if all 5 stages are trying for the same frequency. For such applications, the ability to tune each section, the better component sensitivity, and the reduced loading effects between stages all make active filters a better choice. This assumes that the frequencies involved are well within the capabilities of the active elements, of course.