Is it clear and obvious? I tend to see the line as a bit blurry, as I see both your and
@pemerton 's points. I tend to come back to the following example, which I had always thought fairly straightforward, but it has a lot of opinions (example is 3e because that's how long ago I posted the original):
Lets say the characters are chasing a villain through the streets of a city that one of the PCs is intimately familiar with. The villain has a few minutes head start but the players know where he is likely going.
The PC (intimately familiar with the city) looks at the DM and says "I'm intimately familiar with this city, chances are I know a pretty good shortcut that the villain doesn't."
The DM looks at his map and sees that the villain is going by a direct route with the players unlikely to catch him. Assuming teleportation magic is not at play does the DM a) give the players no option other than to try and catch the villain by directly following him or b) allow the player (assuming he rolled well on a geography check or similar skill roll) to find a previously unknown route (maybe not even on the map) that allows them to catch the villain?
Interestingly, 5e has built a solution to this in. If the PC has the urchin background, he CAN per RAW find a better route and cut off the villain.
My point is, when framing the scene are we really saying the DM has closed off all options other than those they have explicitly stated?