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I'm not exactly sure how changes to ability scores in combat affect abilities that have uses based on the modifier for that that skill.

For example take a bard with a Charisma score of 18 that has already used his 4 Bardic Inspirations. He then draws the Star card from the Deck of Many things. He uses it to increase his Charisma to 20, making his Charisma modifier +5.

As the number of Bardic Inspirations is determined by the Charisma modifier, would its increase to +5 give him an additional use of Bardic Inspiration?

On the other hand, suppose the bard has an Ioun stone taking his Charisma to 20. He uses 4 Inspirations, and then steps in a Antimagic Field, disabling the Ioun stone and reducing his Charisma to 18(+4). Would he have one available Bardic Inspiration or none?

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Whenever your Charisma modifier changes, so does your maximum uses of Bardic Inspiration

Unlike spell slots, you do not have "bardic inspiration slots", you have a maximum that is equal to your Charisma modifier. If your Charisma modifier changes, so does your maximum.

When you try to use Bardic Inspiration you compare your number of uses with your Charisma modifier:

You can use this feature a number of times equal to your Charisma modifier (a minimum of once)

Examples

Following your example 1:

  • The bard has 18 Cha, so can use Bardic Inspiration a maximum of 4 times
  • The bard uses Baridic Inspiration 4 times, they are now at their maximum.
  • The bard draws the Star, and now has 20 Cha, so can use Bardic Inspiration a maximum of 5 times, so they can use Bardic Inspiration 1 more time.

Following your example 2:

  • The bard has 18 Cha, so can use Bardic Inspiration a maximum of 4 times
  • The bard has an Ioun stone taking their Cha to 20, so they can use Bardic Inspiration a maximum of 5 times
  • The bard uses Baridic Inspiration 4 times, they have 1 more usage left.
  • The bard enters an Antimagic Field, disabling the Ioun stone and reducing their Cha to 18, so can use Bardic Inspiration a maximum of 4 times. They are now at their maximum.

Medix2's bonus example:

  • The bard has 18 Cha, so can use Bardic Inspiration a maximum of 4 times
  • The bard has an Ioun stone taking their Cha to 20, so they can use Bardic Inspiration a maximum of 5 times
  • The bard uses Baridic Inspiration 5 times, they are now at their maximum.
  • The bard enters an Antimagic Field, disabling the Ioun stone and reducing their Cha to 18, so can use Bardic Inspiration a maximum of 4 times. They cannot use Bardic Inspiration, since they have used it more times than their maximum. The expended uses still remain in play, only the number of times the ability is "used" is limited by the maximum.
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The Bard's available uses of Inspiration is evaluated constantly and may change at any time if their charisma bonus changes.

The rule for Bardic Inspiration concludes with this information:

You can use this feature a number of times equal to your Charisma modifier (a minimum of once). You regain any expended uses when you finish a long rest.

There is no specified time for the quantity of Bardic Inspiration uses to be evaluated. Without such a limit, it is therefore constantly being evaluated.

If the maximum quantity of uses increases as a result of the bard gaining charisma, s/he gains an additional use right away. Likewise, if the bard loses charisma and a corresponding use of Bardic Inspiration, that reduced maximum is enforced at the moment that happens. This could deprive the bard of unspent use(s) if s/he had at least one use left.

The bard would regain up to whatever the new maximum is after finishing a long rest, provided the change to his/her charisma score is maintained.

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