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I have a home brew Legacy combo deck that can win by about turn 4-5 reliably. If I change the deck a little to Modern it can win probably on turn 6-7. It is weak to burn and has a few counterspells to protect my assets.
My question is, how fast do competitive combo decks typically win in the various formats? By that I mean in Legacy, Modern, Vintage, Standard, and EDH. Is a turn 5 win slow for Legacy?
Is a turn 5 win slow for Legacy?
Yes.
As of the current meta (from May 24th, 2018), Turbo/Dark Depths is the most represented Legacy combo deck, having around 6% of the field. This specific archetype can combo as fast as "Turn 0" with as few as 5 cards (a land that produces green mana, Crop Rotation, Vampire Hexmage and two Lotus Petal). It can consistently combo on turn 2 or 3, so this is usually what a combo deck should aim for.
The next 4 most represented decks are Storm, Show and Tell, Elves and Reanimator, all of which can consistently combo as soon as turn 2, but usually go off on turn 3-4.
Dredge decks don't benefit as much from the 0 and 1 CMC mana rocks, but it can usually go off on turn 2 or 3 with a couple of Bazaar of Baghdads.
The Modern metagame is all over the place, and it's mostly aggro decks, having 54% of the field.
The most represented combo decks are Scapeshift, U/R Storm, GWx Devoted Company and Living End, all of which usually combo on turns 3-6, depending on mana acceleration and card draws.
The current Standard meta has no combo decks. The last time a combo decks had any relevant appearance in standard was during the BFZ-AKH standard, where both Aetherworks Marvel and Saheeli Cat decks worked more in a control role until they could go off. They're considered combo decks because their wincon is a combo, but they were not particularly fast.
Combo decks in duel commander (aka "The French list") are currently much underrepresented. The three most played decks still only represent about 3% of the field.
They're Iname, Death Aspect, which is a control/reanimator deck, and Prossh, Skyraider of Kher, which is a "cast my commander, hit you for 21" kind of deck.
Teferi, Temporal Archmage is represented in both MTGO (9% of the field) and Duel commander (1%) and it is really a control deck that controls the game long enough to cast either Emrakul, the Aeons Torn (in MTGO) or Emrakul, the Promised End (in Duel).
Disclaimer: Every number mentioned in this answer was taken directly from MTGTop8, which really mostly takes MTGO into account when breaking down the metagame, so some of the numbers might be slightly off when talking about physical decks. Also, the information on how long a particular deck takes to combo comes from personal experience playing either with or against said decks and watching tournaments and streams.
I feel like the current answers & comments are a bit too aggressive so here's an attempt to get some sourced answers for this question.
In Legacy, turn 5 with no disruption from the opponent is too slow. There aren't many out-and-out aggressive decks (ironically because doing that gets you killed by other combo decks), but if someone e.g. plays Legacy Humans they can quite reliably do 20 damage on turn 4 while simultaneously presenting disruption in the form of Meddling Mage, Kitesail Freebooter, and especially Thalia, Guardian of Thraben (which decks other than Humans also plays). See this video.
You're competing against other combo decks such as Elves, which will usually go off on turn 3.
By most standards, the Elves mirror match is a nightmare pairing. In game 1, neither player can disrupt the other and, left to its own devices, Elves goldfishes on turn 3 (exactly turn 3) in a very, very large portion of games. Under such conditions, it’s difficult to break serve, and unless something strange happens, the die roll effectively tells you who’s going to win.
Elves is a consistent combo deck in the sense that it doesn't fold to a single piece of disruption and is capable of playing a grindy game. Other, all-in combo decks such as Reanimator can goldfish faster (although a single Force of Will can mean it crumples like a house of cards). It's not so easy to get a source as to how fast Reanimator combos since it's sort of common knowledge, but you can get an indication from this article (note that Reanimator doesn't actually "win" on turn 1, it just puts the game to a state where the opponent cannot realistically come back and is better off conceding to hide information):
Sideboarding in the Dark
What?
Yeah, you read that right. It actually happens quite often that you win on turn 1 on the play and you have no idea what your opponent is playing.
Edit: Hall of Famer Reid Duke wrote an article including this quote on the speed at which combo decks go off in Legacy:
The faster combo decks in Legacy will "go off" on the first turn between ten percent and fifty percent of the time. They are designed to go off on the second turn with incredible consistency, even through a single piece of disruption. This means that if your anti-combo cards are primarily spells that cost one or more mana - like Thoughtseize or Spell Pierce - you’ll be dead in the water when the combo player has a good hand on the play.
In Modern, combo speeds are slower because of the turn 4 rule. When Wizards created the format, they said:
We used two criteria to guide us in choosing what cards to ban. First, we have a rule of thumb about Legacy that we don't like consistent turn-two combination decks, but that turn-three combination decks are okay. We modified that rule for Modern by adding a turn to each side: we are going to allow turn-four combination decks, but not decks that consistently win the game on turn three. We banned enough additional cards that we think such decks no longer exist in this format. Those additional cards are as follows:
(Of course this paragraph also says decks don't reliably combo on turn 1 in Legacy, and Reanimator breaks that rule, but presumably Wizards allows Reanimator in Legacy because of how fragile it is.)
Some decks can combo faster than turn 4 given good draws, but no deck can consistently do it, or they would get banned.
I don't have any sources for Standard & Vintage, but you get the idea: the more powerful the format is, the faster the combo decks. With Legacy decks consistently comboing on turn 3, you would expect Vintage decks to do so on turn 2. Standard decks are the most difficult to classify since the format changes so fast, but you would expect them to take more than 4 turns, i.e. 5-6.
Disclaimer: all this assumes no disruption from opponent. It's quite common for combo decks to decline to combo if they're not under pressure, to play around disruption.
