This is solid advice, also because if you think you’ll be 2 weeks late it is likely that you’ll eventually be 3 weeks late anyway. Delays have a habit of piling up.
I'd actually see that as a positive thing; too much money is wasted on the sunk cost fallacy, and having the guts to cancel a project if it goes too far over time / budget is rare enough.
I would argue that it is impossible to form any conclusions here without knowing what the exact specific situation is, but of course what you described is plausible.