Thats b*llox! You only see the mongrels and crossbreeds that have survived, the others have died young! Anyway, how do you think mongrels and crossbreeds are bred - they are the product of other breeds. For instance if you bred say a Cavalier (heart problems, eye problems, immune system problems etc.) with another breed or crossbreed, it could pass on these problems.
It is more to do with the general upkeep and feeding etc. that decides whether they live longer.
Generally bigger breeds do not live as long as smaller ones, and this is the same for mongrels. A Rottie x GSD will probably live no longer than either purebred Rott or GSD, particularly if it inherits hip problems, intestinal problems, eyes, etc.
A smaller mongrel or cross may well live longer, because it is the product of two breeds that are known to live long lives.
My breed is small (average 12 inches at the shoulder). They live usually into their mid teens, and many have reached 18 or more.