I ran into this error today in SQL Server 2012. It was caused by having promoted a package-level connection manager to project-level and then copying and pasting some data flow components from one package to another in the same project.
By copying the data flow components from one package to another, the SSIS designer also copied a reference to the connection manager, but apparently didn’t catch on to the fact that it was a project-level connection. Thus, it duplicated the connection manager’s key in that package which caused the error described above.
The solution was, after having pasted the data flow components, to remove the newly added package-level connection manager which will make the project-level connection manager show. Then attempt to save again and it should work.
Sven Aelterman 
I was on the way to hanging myself. Thanks.
@Mark: You’re welcome. Glad someone was saved.
Saved me as well, Thanks Sven
I have 200 packages in my solution, how do I know which package has this issue?
Jayant: does the Error List not specify the package?
This was the fix for me today with SSIS 2016. Thanks, sir!
Good to know this is still relevant!
Just saved me a bunch of time—thanks!