It should work.
I have tested with Joomla 1.5.12 where there is an update that require to JMS re-isntall the patches.
In Joomla 1.5.13 there is no patches to install if you come from a Joomla 1.5.12.
I suspect that you have installed Joomla 1.5.13 over an older Joomla version prior 1.5.12 that explain that JMS has detected that something has to be re-installed.
You have not mentioned the list of files that had to be fixed by JMS before you received all the files to re-installed.
I suppose you have a permission issue at least on one file or several files that had to be patched.
If you remember the file(s) that had to be patched before to receive the whole list, this would help you identify the file(s) that has (have) a permission issue.
One possible reason is that you have installed the Joomla 1.5.13 patches with a user that have more privileges than the apaches or IIS user that is used when you ask to install the JMS patches.
Check the owner and group permission of the files.
Perhaps you should set a chmod 777 or try using the FTP Layer to try re-install the pathes.
See tutorial step 15 on how to use the Joomla FTP Layer.
www.jms2win.com/tutorial