Can I use the "import" tool on PHPMyAdmin multiple times? I am moving databases between a couple of shared hosting accounts, and both of them only give access to the MySQL databases through phpmyadmin.

The importing account's upload limit is 50MB. I would like to export the database in parts (by selecting only a few tables to export at a time), using the export tool, then use the import tool more than once to get around the limit. Is this possible?
Yes this is possible, although you need to watch out for the following:

a) Export using extended inserts so that the extended SQL syntax is used which will reduce the size of your SQL files (If you are exporting using PHPMyAdmin select both complete and extended insert options, as it creates complete inserts with extended insert syntax for large tables) 

b) Try using the GZip format which will compress the SQL files and make them smaller (depending on the hosting provider setup this may be an issue so you need to test it out with a small table) 

c) Export the tables in such an order that tables with foreign keys are imported last