Uh, not by default is the best way to answer this question.
It won't. Think of normal filesystems (ext3, reiser, xfs) and partitions (hdb1 hda2...) as seperate drives themselves. You couldn't just have data flow over from one drive to the other without something in place that deals with this. My preferred way is to LVM to do what you are asking.
Using LVM you can connect a bunch of drives together to one device, create 1 partition across the drives, and mount it at a given mount point. However, once that mount point is full, if there is no interaction by the user/admin (to add more space) then it will simply stop writing to that mount point. You'll get a 'disk full' error (depending the app it may word it differently). You'll still be able to write to other mount points, assuming you have other devices/partitions mounted to them (such may be the case with /var or /tmp depending on your setup). With LVM, once your data nears completely filling the disk, you can easily plop in another disk and using lmvtools you can grow the volume (the partition per se) and then the filesystem (xfs_grow for example) and have a bigger device (as it will appear).
RAID is another option.
If all of that is confusing, the easy answer is no. Once the partition is full, it's full.