Take a look at a variety of used motor oil analyses on the board. You'll see both happening.
It depends on the oil and the use.
Run an oil a lot in applications where it shears it, like some motorcycles, and it thins out.
Put it in a car prone to sludge and do a lot short trip driving, and it can thicken.
Also, don't change it and it will thicken.
Oil generally thins initially, over the first few thousand miles. This is from shear forces, fuel dilution and other reasons. Then oils thickens during the remainder of it's use. This is an oxidative process. The time or mileage period varies with the additive package of the oil and the oil's general base stock. Synthetic oils generally this less at first and thicken less over the long run.
Some oils have a more shear resistant viscosity index improvers (costs more). These won't thin much if any. Some oils have a more oxidation resistant base stock and more effective antioxidants in the additive package, plus are less volatile. These won't thicken much.
If an oil thins from shearing and thickens an equal amount form oxidation, is still is no good...both are negatives regarding engine protection.