Ted's not stupid, so why...
...does he ramble on at lunch with his boss and tell the cutesy story about Billy saying "You've lost weight--and it went to your nose!" (paraphrased).
Ted *knew* that his boss was already concerned with his lack of focus on the job; why would he babble on about Billy when this was very clearly an issue? Ted's no idiot.
Lame writing, imo.
That said, one of my favorite parts of the movie is when Ted says, "Shame on you" in refusing his boss's cash offer after said boss fires him. Very powerful acting by the always-terrific Dustin Hoffman!