Most airlines will give you status if you have status with another airline they consider a competitor, in order to remove a barrier to your moving your allegiance to them. This is called "comping" status. Searching for that term here will turn up several threads.

Airlines vary with how readily they will comp status, which other airlines they consider competitors for this purpose, and what levels they will comp to. AA, for example, will never (barring some unusual situations that are unlikely to affect anyone who has to ask the question) comp its top level, even if someone has comparable status elsewhere. You can find specifics for airlines of interest in the forum for that airline, by searching for "comp" and/or posting a question there.

Comping always requires proof of current status, such as a copy of an FF program statement. It can often be helped along by a word from a supportive travel agent or corporate travel coordinator.

This is not excatly honoring your status with another airline. It's giving you status on their airline. Honoring status with another airline is governed by inter-airline agreements. It generally applies within alliances such as OW and *A.

Comping is usually a one-time deal, so choose your moment carefully. You have to requalify for status on your own and are unlikely to get a second comp from the same airline if you don't. This is a bad time of year to get a comp, unless your travel plans are heavily weighted to summer and fall: it's late enough that much of your 2002 mileage-earning potential has passed, but early enough that they're unlikely to comp through 2003.