Yes.  And not just for the inbox, but inbox and sent may be the only folders synchronized by default. 

    From within the android gmail client:
    Press menu
    select more
    select settings
    scroll down and select labels
    change the number of days from which you want messages kept on the phone
    change individual labels (effectively folders) to none, this number of days, or all

NOTE: this answer was written about a now obsolete version of the gmail app, which continues to change in its user interface over time.  The answer *will not* be updated to track that evolution, as there are multiple versions in use at any time which behave differently, and it is ultimately the idea rather than the specific steps which are important.