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Given that every class gets a minimum of 5 ability score improvements each at levels 4, 8, 12, 16, and 19 (Rogue and Fighter get more at other levels), do these ability score improvements occur by character level or by class level?

For example, is it possible to create a character who never once gains a single ability score improvement by reaching level 3 in a class and then switching to a different class repeatedly?

I had originally thought the 5 that every class gets would occur by character level, so for example a sorcerer1/bard3 would earn an ability score improvement for reaching character level 4. But there are others who have expressed the opinion that this is wrong.

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up vote 20 down vote accepted

Ability score improvements are a class feature, and are only gained when the class level grants it. Different classes gain these advances at different levels (the Fighter is the clearest example of this), and there is no unified table of advancement by character level.

It's possible to never gain an advancement by always multiclassing into a new class before reaching 4th level in a class. One effect of this is that "dipping" a class for a single level or two is less obviously useful; considering the extra limits on what is acquired from a new class's first level when multiclassing, this is likely a significant reason for the design, apart from being able to give classes different advancement rates.

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