Short answer. No
Non of the versions are backwards compatible.
Finish your current project in E13 and then upgrade.
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Ann is right. There are no guarantees that your Premiere Elements 13 projects will open in version 14. They usually do, but it's best to keep a copy of the original or risk corrupting it. Or, better, as she advises, finish your work in the version you started it in.
Things are different with Photoshop Elements, since it produces PSD file rathers than project files. You should have no problems opening in PSD files in newer versions of Photoshop Elements.
You are not really asking about backward compatibility as in opening a 14/14.1 project in 13/13.1. Rather you are asking about opening a 13/13.1 project in 14/14.1. That is not backward compatibility in the typical sense. Please read on to see what I mean.
If you are working on a project in Premiere Elements 13/13.1, it is likely that you can open that saved closed 13/13.1 project file (.prel) in Premiere Elements 14/14.1. No guarantee on that, but high probability of success. The caution is this. You cannot open the 14 edited 13 project again in 13. As a safety net, best to work with a copy of the 13/13.1 project when you want to open it in 14/14.1. That latter point deals with "backward compatibility".
Is all this work being done on the same computer? If so, there are considerations to assure that the 13/13.1 project file travels with its source media to 14/14.1.
More on that if that is applicable to your situation.
Finishing projects from an earlier version in a later version is not a guarantee, but to answer your question - it is possible.
Please consider, and do not hesitate to ask questions or ask for clarification on the above.
I am assured, as 3 of you say so, that I am best to finish my project, then possibly upgrade the software.
Many thanks to Ann, Steve and AT.
Thanks for the follow up. Appreciated.
We look forward to the opportunity to be of assistance, if and when needed, as your move forward with your Premiere Elements projects.