Short version:
I've got 17 cousins but have no real contact with any
DS has got 3 full and 4 half cousins but no real contact for years
DH has 3 cousins - has social media contact with one and they message occasionally over a shared hobby
If the family doesn't have people who are prepared to organise bringing the family together or there is any kind of dysfunction then I think cousin relationships suffer.
I had 20 cousins - all on my dad's side. He is one of six (who had 4, 8, (dad -3), 3, 2, 3, children each). It is now 17 as three died early (one as a child, and two as fairly young adults). My mum was an only child. We lived in England and most of them lived in Scotland or further away in England so only ever saw them very occasionally - most were adults/teens when I was born, only three are around the same age. I was facebook friends with three of them but I've come off facebook now so that is that.
My son has three cousins from my husband's side (but hasn't seem them for 8+ years due to dysfunctional family dynamics) and four half cousins from my side (I barely know the half siblings and would have to look up their names, and the cousins are more my generation than my son's).
DH's saw his cousins a bit growing up but only sees them at funerals now but does have some social media contact with one as they share a hobby (doesn't live that close either and almost old enough to be generation above).