I am aware of project organisations that have had problems spending their money, but in such cases the observation about projects needing more people or "time" actually concern volunteers (or "managers" in the business dictionary) who can make sure that money gets spent on useful things, not those who will actually write code or documentation.
Such organisations often don't want to frivolously spend money on things that wouldn't help the project in some way - massive billboard advertising or "Brewster's Millions" endeavours are seen as being irresponsible with other people's money - and thus they can get bogged down in micromanaging grants for worthy causes. Even with a healthy stream of worthy causes being suggested (adding functionality, writing manuals or courses), there's always the concern that such activities won't be sustainable after the paid individuals have finished their work: might such things not just add to the project's workload?
That said, I see a certain amount of benefit in targeted donations where someone might say that they have a specific objective and will pay for it to happen, thus providing the worthy cause and the cash. That isn't so different from a "bounty" - something that Mr Shuttleworth used to offer for various Python projects, as I recall.