Brawl
"Only Nintendo characters will be included in Smash!"
-Snake from Konami's Metal Gear series was one of the first newcomers revealed.
"There will be no cuts!"
-Five characters playable in Melee were cut from the roster.
"The only cuts will be clones!"
-Mewtwo was among those cut, albeit he was planned and had the most work done--or left in the game data--out of all of the unused characters. Additionally, two of Melee's clones--Falco and Ganondorf--returned with a few differences to set them apart from their originals, and Young Link arguably morphed into Toon Link.
Smash 4
"A character from a series with only one game won't be included!"
-Shulk was included, despite there only being one Xenoblade game released. However, Xenoblade X (then known only as "X") was announced--as was the New 3DS port of the original Xenoblade--before Smash 4 released. However, both games would release after Smash 4 did; additionally, Shulk was decided upon to be added in 2012, before either game was made public (whether or not Sakurai had inside knowledge of Xenoblade X at the time is unknown).
"A character can't be both a stage hazard/prop and playable!"
-When Spirit Train was revealed for the 3DS version, it was shown that Toon Link was conducting the train, which some felt meant that Toon Link was out. However, Toon Link would indeed join the game, and it was revealed that if Toon Link (or Link) was fighting, Alfonzo ("Leave it to me!") would take his place in the back. Granted, this role does not interact with the fighters at all, so we still don't know how this rule affect actual interactable stage characters.
"The Lord of the newest Fire Emblem game will always be picked over anyone else from their game!"
-Chrom, the main character and Lord of the newest FE game at the time, was infamously snubbed in favor of the Avatar character Robin, due to Robin presenting a far more unique flavor than Chrom would have.
"Once a character has been cut, they won't come back to the roster!" (I actually did see this.)
-Dr. Mario returned in the base game, with Mewtwo and Roy joining in DLC; all of them were removed in the transition from Melee to Brawl.
"If a character has been in at least two Smash games, then they won't be cut!"
-The Ice Climbers, playable in both Melee and Brawl, were removed, albeit due to hardware limitations with the 3DS and Sakurai's desire to keep the roster the same with both versions of the game.
"No third party company will have more than one character in Smash!"
-Ryu showed up as Capcom's second character after Mega Man.
"A character needs to have their game released on a Nintendo platform to be included in Smash!"
-Cloud from Final Fantasy VII, a game that has never touched a Nintendo platform, was included. As far as Cloud is concerned, he has only has menial appearances on Nintendo platforms (Kingdom Hearts cameos, Theatrhythm, Final Fantasy Explorers).
"A third party character needs to be a notable icon in order to be included!"
-A controversial one, and one based on Sakurai's comments on the subject, even. Regardless, Bayonetta was included, despite not being a cultural icon like the other third party characters; however, her entry was due to the Smash Ballot, where she is stated to have gotten the most votes among "negotiable and realizable" characters. Due to Nintendo's heavy involvement with the IP in recent years, the lines between first party and third party may be blurred for her.