@JeremyECrawford Can Boots of Speed increase your swim or climb movement, or just walking only? Not talking about a Swim _Speed_, just base walking Speed used for swimming
— keithcurtis (@keithcurtis) December 12, 2017
If you lack a swimming/climbing speed, you use your current walking speed to swim/climb but must spend extra movement. The swimming/climbing rules don't shut off increases to your walking speed. #DnD https://t.co/QYkbOO5GvD
— Jeremy Crawford (@JeremyECrawford) December 12, 2017
What about for creatures with climbing speeds? If a Tabaxi wears Boots of Speed, does their climbing speed increase as well?
— Sol K (@SollyDoodle) December 12, 2017
Boots of speed increase your walking speed, not any other speed you might have. #DnD https://t.co/rTDIeaD9gH
— Jeremy Crawford (@JeremyECrawford) December 12, 2017
It is important to point out all the complexities of the “Speed” system. There are different terms which mean different things.
Some abilities, items, et cetera, specifically state “walking speed”. This quite naturally applies to your speed while moving upright on foot. Or they might specify climbing speed, which naturally applies to vertical movement on all four limbs. Or they might specify swimming speed, which naturally applies to movement in a liquid.
But not everything that can increase your rate of movement specifies a certain kind of speed. Some abilities, items, et cetera simply state “speed”. For example, a 5th level Barbarian gains the Fast Movement ability, which increases their “speed” by 10 feet while not wearing heavy armor.
This generic bonus to “speed” applies to all kinds of speed a character possesses. If they are a Tabaxi, they have both a walking speed and a climbing speed of 30 feet, and both of those increase to 40 feet. Lizardfolk have a walking speed and a swimming speed, which both increase. Aarakocra have a walking speed and a flying speed, and both increase. Et cetera.
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Where things get weird is when dealing with effects that don’t actually grant a certain kind of movement speed, but instead state that a certain kind of movement no longer costs you extra movement.
If you are a Tabaxi with Boots of Speed, they give you a bonus to your walking speed, but not to your climbing speed. Your walking speed goes from 30 feet to 40 feet, but your climbing speed of 20 does not increase.
But if, for example, you are a Rogue with the Thief subclass, and you have the Second-Story Work ability, things are different. It doesn’t grant a climbing speed. Instead, normal climbing no longer costs you extra movement.
When you climb normally, without using a climbing speed, you use your walking speed to determine the distance you travel, but at the cost of only achieving 1 feet of movement for every 2 feet of walking speed.
If you have an ability that lets you avoid that cost, you can climb at your full walking speed. And that includes any bonuses to your walking speed you have.
So our Tabaxi Thief in Boots of Speed has a walking speed of 40 feet, and a climbing speed of 20 feet. But they don’t have to use the climbing speed. They can instead use their walking speed, without having the pay half of it as cost. They can climb a full 40 feet using their walking speed, instead of their normal 20 feet while using their climbing speed.
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One final example to recap. Let’s look at a Lizardfolk Barbarian.
Lizardfolks have a walking speed of 30 feet and a swimming speed of 30 feet.
If they take the Athlete feat at 4th level, they do not also gain a climbing speed, but they can now use their walking speed in place of a climbing speed at no extra cost. They have an “effective” climbing speed equal to their walking speed.
At 5th level, they gain the Barbarian class ability, Fast Movement, and their “speed” increases by 10. This applies to all their kinds of speed. They now have a walking speed of 40 feet, and a swimming speed of 40 feet. They still don’t have a climbing speed, but they can use their walking speed for climbing at no cost.
If they find some Boots of Speed, their “walking speed” increases by 10 feet. They now can walk and climb 50 feet, because they can use their walking speed for both forms of movement. But they still swim at only 40 feet, because the bonus does not apply to their swimming speed.
If they take the Mobile feat at 8th level, their “speed” further increases by 10 feet. Again, this applies to all kinds of speed. So now they can walk 60 feet, climb 60 feet using their walking speed, and swim 50 feet.