I think every city does things a little bit differently, but it seems to be relatively common that a person's street address indicates both where on the block they live (at least roughly), and where on the length of the street their block is.
My street number growing up was 5605. People often referred to my block as the "fifty-six hundred block of MyStreet". The block on one side was 5500, and the block on the other side was 5700. There were interruptions whenever the street didn't go through or there was some multi-block thing that got in the way, but the first two numbers gave you context about where on the street the block was.
My particular area of the city advanced the house numbers one at a time, with evens on one side of the street and odds on the other. So my next door neighbors were 5603 and 5607. My neighbor across the street was 5604. Other neighborhoods in the same city advanced house numbers by 4, so 5604 would be next door to 5608. But even on those streets, you still had even numbers on one side and odd numbers on the other.
Where I live now, at least in downtown (where the streets really are on a grid), the block number is related to which numbered street it crosses, similar to what Toots describes. We have numbered avenues running east/west through a big chunk of the city that get smaller the further north you go. As you get away from the city center, the streets become more twisty, so the avenues don't always go through or mean much of anything. But in the parts of the city that are on the main grid, the avenues dictate the block numbers for the named streets that cross them.