Trains mostly "make up time" by consuming schedule padding (officially known as "recovery time"). They don't speed. There is padding only at major stations and it is a known quantity. So the estimates on the website should be fairly accurate, especially as the train gets within 100 or so miles. With that said, late trains will depart as soon as possible and will reduce station dwell time to the minimum necessary to get station work done. So at a major station, if the train has a scheduled dwell of, say, 50 minutes as a Denver, and they can get the work done (deboarding passengers, garbage collection, watering, fueling, inspections, crew change, boarding passengers are typical at major stops) in 25, they'll be out of there in 25. The station dwells are padded, too.
If a train was showing 55 minutes late, personally I would arrive at the station about the time I would have anyway. If it is showing 4 hours, I would keep tracking it, and show up to the station based on the trains progress. I would not try to cut it too close, as they will do everything possible to make up time, though speeding is not one of those possible things.