Not exactly. To quote Merriam-Webster, a "Catch-22" is a difficult situation for which there is no easy or possible solution. -For example: "no work unless you have an agent, no agent unless you've worked." It gives the idea that there is no simple solution to the problem. Whereas, "That doesn't add up" basically means something isn't right here. -For example: "I gave you $5 for the $2 candy, and you only came back with $1." --That example literally doesn't "add up". Just remember that when something doesn't add up, there is usually some details missing.

"Catch-22" is the title of a very good novel by the American writer Joseph Heller, and is the origin of the phrase. It's a bitterly satirical, black humor novel about bomber crews in World War II. A "catch" means a trick or a deception that prevents you from getting what you expected. "This camera only costs $50." "That sounds great, what's the catch?" "We only sell it with a case, and the case costs $200." In the novel, one explanation of Catch-22 is: "There was only one catch and that was Catch-22, which specified that a concern for one's safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions." A "Catch-22 situation" is one that can't be solved because of a logical contradiction. For example, you need a loan from a bank, but to get one you need to be able to prove to them that you're able to pay it back--but if you were able to pay it back, it would mean you didn't need the loan.

Not really. A Catch-22 situation is a paradox, while "that doesn't add up" means that the explanation or reason for something doesn't make sense.

"A Catch-22 situation" has a very clear meaning: http://www.phrases.org.uk/meanings/catch-22.html