Our landlord has a "no pets" policy, and this is the first time I've signed a lease with such a stipulation and no rider for our cats. However, we were assured that they do not mind our having the cats (I never mention the snake as he is undetectable), and several of our neighbors with the same lease have dogs who are not even well-behaved.
New York City is odd though: when we first got here we found out that most of the places with a "no pets" policy have no issue with tenants having cats. There is about 1 kind of lease for everyone in the city, which includes crap like being responsible for carpeting 80% of the floor and whatnot, and the lease and reality are completely removed from each other here. Also there is a law that if a landlord is aware of your pet and does nothing to make you relinquish them within a few years, then you get to keep the pet regardless of what the lease says.
I've also lived elsewhere in the country, and while we are always very specific about having cats, we happily sign leases that say "no waterbeds" (which is most of them), and we've had a waterbed in every apartment we've had. Sometimes the landlord or their representative has come in for a repair or something, and no one has ever mentioned it. These have always been hard-sided waterbeds, by the way, too.
So, in contrast with the other folks here, I will say that while it probably could be a problem later on, from my experience it is not likely given that you were up-front about the situation with them and they said they were fine with it.
(Somehow I always imagined you with a turtle... :shrugs: )
-Sean