Only 3 Basis Vectors for $\mathbb{R}^4$. I've heard that for $\mathbb{R}^n$, there can only be $n$ number of basis vectors for it. Is that really the case though? Or am I having a misconception somewhere.

Let's take the example of $\mathbb{R}^4$. It seems that I can pick only 3 basis vectors $(1,1,0, 0)$, $(0,1,1,0)$ and $(0,0,1,1)$.

Firstly, these basis vectors are linearly independent. For example, no linear combination of $(0,1,1,0)$ and $(0,0,1,1)$ yields $(1,1,0,0)$. This is because these linear combinations will still have a zero as the first number in the vector.

Secondly, it (seems) that these 3 vectors span $\mathbb{R}^4$.
Can $(0,0,0,1)$ be written as a linear combination of these vectors?
The first statement of your question is absolutely correct that's what is referred to as the dimension of the vector space. 

But coming to the example, we can see that your assumed basis vector (1,1,0,0), (0,1,1,0), and (0,0,1,1) does not span the entire vector space. If you add a vector (1,0,0,0), this completes your basis vector which spans the entire 4 dimensional vector space.