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
Pick your account (e.g. [email protected])
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.
