Can a Mac Pro be upgraded from a 5150 CPU to a 5365?

Yes, you can but only with Intel Xeon E53xx, X53xx or L53xx series processors that have a LGA 771 socket.

Although one word of warning, the supported OS upgrade life time of the Mac Pro 1,1 and 2,1 looks like its coming to an end. Apple as we have seen in the news is prepping for Mountain Lion, and dropping support for all macs that lack a 64bit EFI. So it appears highly likely that Mac OS X 10.7 is the last OS you will be able to run on these Mac Pros, unfortunately. Be aware of that when prepping for your Mac Pro 1,1 and Mac Pro 2,1 upgrade plans, especially if keeping current is a priority of yours.

That said, now on to the upgrades:

The Mac Pro original Upgrade process

Here are processor upgrades that people have completed and confirmed in Geekbench as working, see their results below.

MacPros using L53xx Xeon processors.

MacPros using E53xx Xeon processors.

MacPros using X53xx Xeon processors.

Yep. Anand even did it when the Mac Pro first came out. I've seen other reports too.