The system does not want the ash level to go over a certain percentage.
That's soot, not ash. Ash level can only increase, never decrease, as that's the part that remains in the DPF after the it has been regenerated. If a regeneration completes, soot load is considered to have returned to 0%. However, ash levels never decrease, and when they reach a certain limit, the DPF is considered full/blocked and MUST be replaced. There's no other remedy for this.
Well, the DPF can be theoretically cleaned or even deleted, but this is not official repair method, and involves essentially the same process as the replacement of the DPF, except of course not every part of the DPF will be actually put back on or put back on new.
All is not lost at that point, the system will still try to initiate a burn. A long highway trip is perfect and the burn is started. But, the computer monitors the burn temperature. The more ash you have to burn, the hotter the burn. Once you hit a certain temperature, the system will shut down the burn.
It's not the temperature that stops regeneration - even though there might be fail safes for temperature, too. But generally a DPF regeneration, once initiated, should never reach temperatures that trigger such a shutdown - unless there's something wrong with the car. Instead there are two other things that can prevent or stop regeneration.
One of them is, that the cars cruising speed or the engine speed falls below a certain threshold (30-60 km/h, 1500-2000 RPM, depending on model), that's required for the regeneration. If this happens, this will not trigger a fatal condition per se, because the regeneration will be resumed when the car has speed up to required cruising speed again. The only problem that can arise from interrupted regenerations is, that soot still gets accumulated until it can be resumed, and trigger the other condition explained below.
The other condition that will prevent from the regeneration being initiated (or resumed) in the first place is, when the _soot_ levels are beyond a certain threshold. This might be 40 to 80% depending in the actual DPF. The regeneration will not be initiated because the burn temperatures would be too high in this case, and it could possibly result in the car catching fire or taking heat damage.
If this happens, but soot level is still below a certain limit (usually <75-80%) a manual regeneration can be initiated by service personnel through VAG-COM. Then they can monitor the process and abort if needed. However, if soot level goes above said limit, regeneration will not be allowed even this way, and the DPF will have to be replaced.