I work for a small institution affiliated with a large university --
When we took placement of Gavin, they offered no family leave... only unpaid leave through FMLA. Women who give birth are entitled to use a few weeks of sick leave / short-term disability during their maternity leave, but otherwise all leave was either unpaid or vacation time. While I was on leave with Gavin (I took sixteen weeks -- most was paid, because I saved my vacation time), the institution created a family leave policy. I believe everyone now receives one week at full pay, then up to four weeks of additional time (at a certain percentage of their salary, depending on time with the institution). The family leave can be used to care for children (however they enter one's family), a sick relative, an agin parent, etc.
I don't know anyone who has received a significant amount of paid time off -- i.e., not disability pay, sick time, vacation time.
I was also able to get a small "grant" to cover adoption costs from my employer. But I really had to dig around to discover that this benefit existed... it was offered through the university, and the benefits staff at my institution weren't even aware it existed (until I told them about it). So dig around a bit...