Let $X=(V,E)$ be a finite, connected, regular graph with diameter $D$. Is it true that, for every $x\in V$, there exists $y\in V$ such that $d(x,y)=D$? (the answer is clearly yes if $X$ is vertex-transitive).
The diameter is 8, but 1 is centered with at most 5 as distance to every other.
Imagine taking a long thin sausage, and drawing four lines from one end to the other end, so that each end of the sausage looks like a cross shape. Then draw a huge number of circles around the sausage.
We've constructed a finite, connected, 4-regular graph. But the diameter is the distance between the endpoints, and the things in the middle of the sausage are genuinely in the middle.
Does that make any sense at all as a counterexample?
A similar construction (where the end of the sausage looks like a hash-sign) gives a bipartite counterexample.