Most poker rooms have the same betting rules & betting rounds in the following format. Whether limit or no-limit, tournament or cash game.
Some card rooms allow no maximum bets and raises that can be made as only if it is only two people involved. The reason for this rule is to limit any damage that can be done by two players working together. Otherwise known as colluding which is highly against the rules. The rooms that allow unlimited raises are in California and Las Vegas. This is a great betting rule for the professional as it gives the ability to get way more value for a player that is making a mistake by not noticing that there is a possible straight flush for example and they have the ace high flush.
I have never known of a Texas Holdem Poker game that does not allow to check and then raise a player that bets. Also known as a "check raise" or "check-and-raise".
Most poker rooms consider a verbal declaration such as "check", "call" or "raise" to be binding only if done in turn. In other words only if it is that players turn. Acting out of turn is always frowned upon though as it gives other players information they shouldn't have yet. And therefor some rooms consider it binding to declare any action whether it is that players turn or not.
New players to the game can sometimes get a break on this type of stuff and even veteran players too. As long as it was a mistake and not intentional.. for example if that player could not see the player before them had cards because that player had them not easily visible behind a stack of chips etc. Which would make it the other players fault. In such instances it is the floor person who has the final say.