The latest version of Sourcetree for Mac as of 19 September 2018 is 2.7.6. You can see in the Sourcetree 2.7.6 changelog that Keychain issues have plagued the Mac application for years.
If I allow Sourcetree to save the remote password (e.g. Github, Bitbucket) in the Keychain, all attempts to connect to that service fail. If I then check the account settings in Sourcetree, the account password has changed to the Keychain password (you can only tell that if they have different lengths).
Sourcetree > Preferences... > Accounts > Edit... > (Contents of "Password:" field)
The only workaround I know so far is to delete the Sourcetree entry for that Git service from the Keychain, and prevent Sourcetree from saving the password every time it asks. Unfortunately, this means I have to reenter the password every time, and untick "Store Password In Keychain".
I expect the only solution is to wait for an update that fixes the issue, but this question is to provide a canonical reference.