For the slinger to operate, it must dip into the oil! At too steep an angle, the slinger may be whipping up fumes and no oil!
At too steep an angle, the intake of the oil pump may be exposed sucking fumes instead of oil!
With the oil filled to the correct level (yet another reason to check the oil every time you mow!), the oil starvation angle should be greater than anything you are likely to safely encounter unless you have a very hilly lawn. However, if you do mow steep slopes for more than a few seconds (e.g., to turn around), it would be worth determining if this could be a problem for your engine.
Two stroke engines do not have this problem since the oil is mixed with the gasoline. As long as the fuel feed is working, the engine should be happy and the mower will mow! However, I do not like 2 stroke engines because of their generally higher production of smoke and pollution.
(From: Walt Conner (firstname.lastname@example.org).)
Oil starvation can happen. If you must mow on a very steep slope, look to see which side the valve cover plate is on. This side of the engine will also have a semi-circular bulge in the crankcase while the opposite side will be pretty well flat. The side with the bulge will have the oil slinger located below the bulge. Keeping this side down slope should keep you in oil anywhere you can stand.
When failure occurs, usually the connecting rod is first to go, sometimes the top main bearing seizes. Either occurrence usually means it's time for a new mower.
For really hilly mowing, you may want to consider a model that uses a 2 stroke engine such as Lawn Boy.