Can one eigenvalue for a matrix produce more than one linearly independent eigenvector?

There is a theorem in my textbook that says, "if v1,...,vr are eigenvectors that correspond to distinct eigenvalues s1,...,sr then the set {v1,...,vr} is linearly independent"

Does this mean that a matrix can only have as many linearly independent eigenvectors as distinct eigenvalues? In other words does the implication go the other way as well (making the set of lin. ind. eigenvectors bigger makes the list of distinct eigenvalues bigger too)?

One of the questions on a practice exam is, "This matrix (it is 3 by 3) has two distinct eigenvalues s1=2 and s2=3. Determine whether A is diagonalisable and explain how you arrived at your answer."

I didn't think one eigenvalue could produce more than one linearly independent eigenvector, so it seems obvious from the definition of diagonalisable, "A nXn matrix A is diagonalisable iff A has n lin. ind. eigenvectors" so the question is too easy, would take two lines quoting the theorem and definition of diagonalisable, but there is a whole page worth of room ... plus it gives you the whole matrix which you wouldn't need to answer the question, so I know I am misunderstanding something.

The matrix in the question is {[3, -1, -3]T, [-1, 1, -7]T, [1 0 4]T} if that helps at all.

Thanks in advance for your replies, my exam is on wednesday :-/