If you clone a Test Suite from one project to another it works, but if you try to clone any more the clones have the correct name but the contents are a copy of the first Test Suite that was cloned.
The workaround is to clone a Test Suite, save the project, exit SoapUI, start SoapUI and clone the next Test Suite, repeat.
Using:
SoapUI 5.2.1
JRE (build 1.8.0_71-b15)
Windows 7 Enterprise SP1