Yes, that's possible.
Information on how to convert from phpBB2 to phpBB3 is here
Basically, you must have both phpBB2 and phpBB3 installed on the same server. After you have finished installing phpBB3 you will have the option to convert your phpBB2 board using the built in converter. The information it asks you for is related to the phpBB2 installation and can be found in its config.php file on the server. All of your users, posts, forums, PMs etc are converted. However, styles and MODs are not (with the exception of the birthday and attachment MODs which are supported by the converter). The phpBB2 database is not converted per se
, it is merely read and converted/moved to the new phpBB3 database, so if it goes wrong your old board will not be affected at all. At the end of the conversion, once you’re happy it’s all working, your phpBB2 database and files can be backed up and removed if you so wish. They are not connected to the new installation in any way.