Well, since you want a live application database to have no CHECKDB faults, and the recommended way to remove faults is to restore... Of course you can rename the "broken" database to stand next to the restored one, and usually you can get some or all of your data out of it. And ideally you arrange backup so that losing the live database is as un-painful as you can make it.

Also of course, by default the restored database uses the previous file names and paths. If you want something else, it's tricky.

I wrote a stored procedure to detach and rename or copy data files as well as (instead of) renaming a database, to get it out of the way of the restored database for instance, but it doesn't work on SQL Server 2005 yet - use of INFORMATION_SCHEMA.SCHEMATA is one thing wrong with it for 2005, I haven't found the others.