No, that's definitely not true. As I said, infinite cardinals (like the cardinality of the naturals ℵ₀) are defined to be equivalence classes of sets that can be placed in bijection with one another. Whenever you have infinite sets that can't be placed in bijection, they represent different cardinals. The set of functions f : X --> X has cardinality 2^X too, so e.g. there are more real-valued functions of real numbers than there are real numbers. You can use this technique to get an infinite sequence of distinct cardinals (via Cantor's theorem, which has a simple constructive proof). And once you have all of those, you can take their (infinite) union to get yet another greater cardinal, and continue that way. There are in fact more cardinalities that can be obtained in this way than we could fit into a set-- the (infinite) number of infinite cardinals is too big to be an infinite cardinal.
You might be thinking of the generalized continuum hypothesis that says that there are no more cardinal numbers in between the cardinalities of power sets, i.e. that ℵ₁ = 2^(ℵ₀), ℵ₂ = 2^(ℵ₁), and so on.