I think if you're trying to get from, say, Roaring Meg to the Lister Hospital, it would be easy to conclude that the network didn't connect the two, because several tries could run you into the town centre with no clear onward path, as described above. The housing areas are famous for not maintaining the standards, but the Vardon Road area, for example, is one where any attempt to use the paths is likely to leave a non-resident of that area lost in someone's front garden with nowhere clear to go. Nor does the network go north-south around Vardon Road (to stick to the same example) in the same way that the road network does.
So, technically, yes, it connects most areas, but not in ways that are universally easy to find, and quite often only if you're coming from the right direction. And it doesn't always connect well out of town - coming in from, say, Little Wymondley is a bit of a roundabout route, especially if you're used to the idea that the main roads provide the principle transport corridors. But the town is clearly better with than without its paths.