Okay, this one is a little tricky to answer, so I'll do it in two sections.
YES-Adding multiple saves from different games
YES and NO-Adding multiple saves from the same game.
YES- If the game has separate slots to use. For example, if it has SweetGame Slot 00 then SweetGame Slot 01 then SweetGame Slot 02.
NO- If the game has only a single slot to use.
In that case, just leave the game on, pop cheevos, then pull out MU(NOT HDD!!!!), change save, pull MU from Computer, Put into 360, load and you're done. If you don't have an MU, pick one up from Wal-Mart. They are around $25 and will make everything much faster. Also you need the transfer cable URL is here:
Xbox 360 USB Memory Card Transfer cable For PC to MU | Xbox 360 | Shop | Xploder Online