$$ A = \left[\begin{array}{cccc}0& 1& 1\\ 1& 2& 3\\ 2& 0& 2\end{array}\right]$$
Clearly first and second columns are linearly independent. The third column is the sum of the first two columns.
The matrix rank is 2. Is it because the sum of 2 linearly independent columns would not count as a linearly independent column?
If so, what about the sum of 3 or more linearly independent columns?
Yes; the rank is 2, which is the number of lin indept columns.
If you had a $4 \times 4$ matrix and the 4th column was a linear combination of the first three, then the rank would be at most 3. (I say "at most" because the third col, for instance, might be twice the first col, hence at most the first and second col would be independent.)
I wouldn't use the phrasing that "the sum of two linearly independent columns would not count as a linearly independent column". This suggests there's some magical property, "linearly independence", that applies to each column individually. But it's a question that can only be asked about collections of vectors (here, columns thought of as vectors).
So, if you call the columns $\vec{c_1}, \vec{c_2}$, and $\vec{c_3}$, then no, the set $\{\vec{c_1}, \vec{c_2}, \vec{c_1} + \vec{c_2}\}$ won't be linearly independent, as it's a linear combination of $\vec{c_1}$ and $\vec{c_2}$.
