My personal observations:
There are many different styles based upon which professors search for doctoral candidates, how they prefer to get in touch, how picky they are when choosing doctoral candidates1, and, in general, how much effort they tend to spend to personally guide doctoral candidates in different phases of their projects.
Yet, one thing at least I have almost never witnessed is that a suitable doctoral candidate as well as funding for them is readily available, but the professor decides against accepting them for merely organisational reasons (such as "having too many people to guide at the moment"). Reasons for this may be, but are probably not limited to:
- The funding is time-bound. With funding that is available in specific amounts per specific timespans, you cannot "save" the money for later. If it is not used now, it is gone.
- Even worse, the opportunity might be gone in the future. If significant amounts of funding are not used, this can create the impression with some administrations that the respective money is not required and thus it will not be granted again in the next year.
- Another, unrelated, but common reason is professors have to coordinate a lot more than their personal guidance schedule. Chances are that an open position for a doctoral candidate is a part of a larger project. People in other institutes or other universities or organisations are expecting to start working on the project at a fixed date, and the open position (or even several at a time) need to be filled by then in order for the team to fulfil its contractual duties.
1: This does not mean that anyone will be accepted, but professors have different stances as to how "exceptional" their applicants need to be and how much "shaping of their researcher personality" still has to happen.