Going past your oil change interval is never a good thing for your engine because when you do not change your engine oil often enough the oil gets dirty and contaminated.
The oil when contaminated and dirty loses it's lubrication properties overtime which then causes wear on the engine bearings and other components in the engine such as the oil pump.
The engine needs proper lubrication through the crankshaft to lubricate the bearings in the engine that keep the piston rods moving and the crankshaft moving.
Eventually the crankshaft and rod bearings will seize up or the bearing will spin causing a spun rod bearing which will then destroy the crankshaft as well.
So always change your engine oil at either 3,000 miles or 6,000 miles to keep the engine working properly as the engine oil is very vital to a properly running and long lasting engine.
Not changing the engine oil or running low on engine oil is a common cause of engine failure at low miles.
Engines can last a long time as long as you take proper care of it.
If your engine is high mileage then it would be a good idea to switch to high mileage synthetic engine oil and you can then change the engine oil every 7,000 miles or so but usually it's still a good idea to change the engine oil at 6,000 miles.
If you notice the engine oil becoming dirty and black then it's usually time to change the engine oil and you should change the engine oil filter as well.