From what I see, I wold first try to check if this is related to certain tests, specifically Get-JiraIssue.Tests or/and Get-JiraGroupMember.Tests.ps1. Can you try to remove them both and run build several times in a row?
Sorry for delay. Will look into this today.
It is difficult to say anything in this situation without putting hands on a repro. Should we be able to reproduce the problem by forking you repository and creating AppVeyor project?
I am asking because I see you have some secure variables in your YAML. They obviously will not be decoded in our fork build, and I do not know if this will affect our ability to get a repro.
I was able to stabilize the build for ubuntu by using our google compute engine build cloud. You can see the builds here starting from build 2.7.14 and on every build is the same - failing due to the same 5 tests.
This leads me to believe that perhaps your build was non-determinisically exhausting some resources of the build machine. I have enabled google compute engine cloud for your account. Add this environment variable to your yaml and see if it makes a difference for you.
Note that we don't keep google compute engine worker 'pre-heated' in the same way we do other build machines so there will be a longer lag for the build to kick off.