Do pilots have the authority to fly faster if their plane departed late to make up the lost time?

Yes, we have the authority to fly faster, unless ATC can not accommodate a faster climb, cruise or descent airspeed.

A little bit about aircraft design. Airplanes are designed to cruise most economically or efficiently just below their maximum designed airspeed. This is typically the speed that was used to plan the flight schedule by the airline.

Airliners are not like a Porsche 911’s that cruise at 70mph, but can do 190mph if they're in a big hurry to get there.

We also do not cruise by setting a True Air Speed. We climb out at an optimal Indicated Airspeed until we intercept our planned Mach Number. This generally occurs between 27,000 and 33,000 feet, depending on aircraft types. We then cruise at a speed which is actually a ratio of our speed to the speed of sound, called a Mach number.

Mach 1 is the speed of sound. The speed of sound varies according to the temperature and density of the medium the sound wave is propagating through. At lower densities and temperatures sound travels more slowly, thus the speed of sound is faster at sea level than at 35,000 feet. At 35,000 feet it's about 700 MPH.

A Boeing 767 has a maximum design cruise speed of Mach .85, meaning it can fly at 85% of the local speed of sound (speed of sound decreases at lower temperatures) before it hits the “Red Line” or Mmo (Mach max operating) speed. We also call it the “Barberpole”. Every airplane is the same in this regard. A 737 is a little slower, w 787 or a 777 is a bit faster, but their normal cruising speeds are all just below their maximum design speeds.

For example, a 767 will cruise comfortably at speeds of no more than Mach .82-.83, and a typical filed cruising speed is typically between Mach .80-.81. The difference between your typical filed cruising speeds and maximum speed is about .02 Mach, which is about 14 miles per hour. If your filed cruise speed is at about 550 mph, which roughly corresponds to Mach .80, then speeding up to Mach .82 yields a True Airspeed of 564 MPH.

Over the duration of a 5 hour Transcontinental flight, this extra 14 MPH will be only put you 70 miles closer to your destination airport than where you would have been had you flown at the filed Mach .80.

It will only take about 8 minutes of flight time to cover those 70 miles by an airplane that took off at the same time and followed you. Then you might get vectored or slowed down more than the guy behind you flying at the flight plan speeds, and they could possibly arrive at the same time as you anyway.

To save those whole 8 minutes, I'd burn an extra 1–1/2–2 tons of fuel, which may negate the profit I might have made on the flight, and have less fuel in reserve if I need to divert because of unforecast significant weather or a closed filed destination airport.

Any story of a pilot who sped up to make up lost time and succeeded is fiction, or was just a delightful and lucky coincidence. Yes, the pilot will say that, but he or she is only trying to make you think they are on your side. They know we can only can't save but a few minutes at best, and run low on fuel at worst. If we did save a lot of time, most likely the tailwinds were stronger than forecast, and we got no delays during the arrival phase of flight.

In summary, we can save very little time on a long flight, and none on a shorter flight of under 3 hours.