My apologies. I am the poster of this question but I struggled with having two accounts. The accepted answer does not seem to solve my issue after all.
So every time when I want to push updates to Bitbucket via Sourcetree I'm getting the following error:
git@bitbucket.org: Permission denied (publickey).
fatal: Could not read from remote repository.Please make sure you have the correct access rights
and the repository exists.
git@bitbucket.org: Permission denied (publickey).
fatal: Could not read from remote repository.
To solve this error I can delete the ssh key from my computer (Mac Pro) and my Bitbucket account and create a new one. Then I can push to remote until I try again a couple of days later. Then I have to replace the ssh keys again.
Since posting the other ticket I noticed this issue also happens when Pulling from remote.
Also the solution which I accepted as correct answer (running "ssh-add -K ~/.ssh/<git_key_name>" in Terminal) seems to work one time only. The next time I want to push to or pull from remote I need to do it again.
I also figured out this issue only happens on one of my computers (Mac Pro), not on my laptop.