I think you have the right idea but the wrong terminology. A n qubit system requires amplitudes (probabilities) for all 2^n possible basis states. For example, the representation we can give to a 2-qubit system is just a probability (technically an amplitude) associated with each of states 00, 01, 10, and 11. It's not really fancy or hard to see how this increases exponentially.
Here's an example with classical randomness (and has a close quantum analogue). If you have two closed boxes with coins in them and shake both around so that the coins flip around in the in such a way that they land 50/50 on each side, you could represent that state of the system as [1/4, 1/4, 1/4, 1/4], that is, each possible state of the system of two coins (heads head, heads tail, tail head, and tail tail), has 1/4 chance of being observed when you open the boxes.
Note that entanglement refers to the idea that two bits can be correlated. That is, the state of one is correlated with the state of another in some sense. In the example I just gave, the coins are not entangled, since the observation of one of the coins does not affect the probability of what the other coin could be. In mathematics we say this is possible if the state of the 2-coin system [1/4 1/4 1/4 1/4] can be written as a tensor product of the constituent 1-coin systems [1/2 1/2] and [1/2 1/2], which it can (entanglement refers to a state where this factorization isn't possible).
That said, we can achieve entanglement with the coins too, by somehow getting a state vector like [1/2 0 0 1/2] (50% chance that both coins are heads and 50% chance both are tails), which cannot be written as a tensor product of two 1-coin states. Moreover, if you open one box and see its a heads coin, you know the other box must also have a head coin. That's what is meant by 'correlation'. To product such a state of the coin boxes, I guess you would need a third party or something to put the two coins in the boxes and delivering them to you.
My conclusion is, entanglement doesn't have to do too much with the exponential size of representation n-qubit states, and I hope I gave a good example of what entanglement kind of really means. While (randomized) classical analogues to quantum effects are not really perfect, I think they demonstrate these effects more understandably.
Parent is correct — entanglement is exactly due to the fact that there exist more vectors in Hilbert space than tensor products (~ tuples) of unit vectors.
