I partitioned my computer to have one 950 gb partition and one 50 gb partition. I'm trying to backup my computer to the 50 gb partition, but once it hits like 30 gb free hard drive space it says there is no more room for backup. I need to know if this is because of the fact its the hard drive on the computer, or something else, because I want to buy an external hard drive, and I don't want the same thing happening.

Try making the partition a little biger! Manage within windows 7 allows you to do it no problem!

It won't let me resize the partition I created in Windows 7. It will only let me shrink the C: drive, but not merge or increase the second drive

In this case, you can extend the partition that you want to copy data to. You have already known that Windows 7 can not let you extend the partition, so, you can choose 3rd party software like Partition Assistant to finish the extend partition operation, it has free ware as Partition Assistant Home Edition, also can do well, have a try, I always use it to extend my partition to maximize my computer performance.

What you need to do is shrink the partition that windows is installed on then expand the partition you want to backup to. This does work, Iv'e done it myself!
Just a thought, I am using windows 7 ultimate 64bit, it's possible that lesser windows 7 versions (home for example) don't support resizing. I just don't know for sure!