Apparently I have changed my password on our BitBucket server, but had forgotten to update the Keychain password SourceTree uses. I tried to do a Pull and got these results:

git -c diff.mnemonicprefix=false -c core.quotepath=false -c credential.helper=sourcetree fetch origin
fatal: Authentication failed for 'URI_REDACTED'
fatal: Authentication failed for 'URI_REDACTED'
fatal: Authentication failed for 'URI_REDACTED'
fatal: Authentication failed for 'URI_REDACTED'
Completed with errors, see above

The value of URI_REDACTED is correct and the same on each line.

SourceTree should not retry a connection after authentication failure, as there’s no way a retry would ever succeed. Besides the “do the right thing” principle, there is also a practical problem with this behavior: the BitBucket server detected multiple failed authentication attempts and locked me out, forcing me to go to the Web site and log in again following the “locked out” method (complete with CAPTCHA).

Software:
SourceTree-Beta Version 2.6 (117)
Mac OS X 10.12.5 “Sierra” (16F73)