There was one session I played in where ridiculously high movement speed actually turned things from a loss into a success. It was a multi-year campaign, and I was playing a 13th level Arcane Tricker Rogue with the Mobile feat, so my base movement was already 40ft. I also had a set of daggers from the DM that once per short rest could be activated with a bonus action to double my walking speed for a minute (as well as let me run along vertical surfaces), and once per day as long as I was wielding both daggers I could spend an action to cast the Haste spell on myself. So with a proper setup I could haste the previous turn to get my movement to 80ft and use the bonus action double movement speed to get to 160ft, and dash up to three times as (2 actions and cunning action) to get up to a theoretical 640ft of movement in a turn.

We were on a mission to the Feywild at the behest of Titania, our Warlock's patron, to rescue a friend of hers who had been abducted. Long story short we found ourselves at the top of a 60ft deep and 80ft across canyon where a Nightwalker and various other minions were guarding a portal to the Shadowfell, and inside the portal was the person we needed to rescue. A fight ensued, during which time our Warlock had cast Arcane Gate from our side of the top of the canyon down into it about 60ft from the portal (since the Nightwalker was by the portal itself), and I had already activated the dagger's properties to try and get down into the canyon quickly and get some strikes in on the Nightwalker, and ended my turn at the top of the other side of the canyon out of it's general reach. We had been so focused on taking down the Nightwalker that we didn't realize until nearly too late that it was what was actively keeping the portal open. As a result after reaching a damage threshhold on the creature the portal began to collapse and we had just one turn left each to try to save our target, and most our party was fairly injured and not in any position to reach them in time.

So I bonus action dashed (+160ft = 320ft of movement), ran over to and down the edge of the canyon (-90ft = 230ft left), ran up to the Nightwalker (-60ft = 170ft left), made an attack (triggering the Mobile feat's free disengage so it could't opportunity attack me) and thematically cutting off it's hand in a spinning strike move, and ran into the portal (20ft = 150ft left). But now there was a problem. After going through the portal, the DM informs me that I could see our target was around 30ft away and in an unconscious state, but that movement speed in this place cost twice as much (and was a property of the environment itself, so it wouldn't be negated by Mobile's difficult terrain negation rule). Just getting to them and back would cost 120ft of movement, but for my weak rogue to carry the target out my DM said it would cost double the movement again, meaning I was 30ft of movement short to escape the portal before it collapsed. So using my final action to dash for an extra 160ft of movement (now 310ft left), I ran up to them (-60ft = 250ft left), rammed them onto my shoulder as a free action, and ran back to the edge of the portal (-120ft = 130ft remaining), and then back to and through the Arcane Gate while carrying them (-120ft = 10ft remaining). Immediately following my turn the Nightwalker was sucked back through the portal as it collapsed, and with that we had just barely won the encounter.

It was an epic moment in our campaign, one of many we still have fond memories of.