Is it actually true, or just a trope? Anyone in a position to manage hundreds of millions worth of projects is smart enough to know that some projects will run under budget.

I work as a federal contractor. It's very true (epistemic status: my managers and project leads tell me as much and I act accordingly, I don't deal with it directly nor understand the bureaucratic larger picture). You will not get funding from Department X again if you ask for more money on a project than you wind up spending. Now, is that the sin of overquoting, or the virtue of overdelivering? For some reason, every agency treats it as the former, and I haven't the foggiest idea why. My coworkers acknowledge how stupid and perverse of an incentive it is, yet treat it like a fundamental force of nature.

Most solutions to this problem are essentially what the OP recognized as nakedly illegal---that is, overreporting productive hours---but most contractors are savvy enough to do it in less auditable and more positively regarded ways, such as stretching out timelines (four 20-hour work weeks raise fewer flags than one 80-hour week), adding more chefs than the kitchen calls for, or funding unnecessary little side projects. Straight-up tampering with timecards is an impatient and dangerous way of achieving (IMO) the same wasteful evil as happens everywhere else in the public and private sector.

I work with people who are well smart enough to know that.

It’s also still a reasonable question to ask “well, last year we budgeted $15M and you got acceptable results while spending only $14M; perhaps you only need $14M/yr…” And despite its reasonableness, many people would prefer to oversee a $15M/yr budget.

The Dutch University of Delft systematically 'maximized' grants and shuffled the money between projects, according to investigative journalists of the NRC[1].

1 https://www.nrc.nl/nieuws/2024/02/15/het-subsidiepotje-moet-...