This question isn't just about Half Elves specifically, but playing as a Half Elf Bard this came to mind. I know that when you reach levels, such as level 4, that you cannot raise an ability score over 20. But does this mean a base score that isn't taking into account racial bonuses? In which case you could get a Half Elf to have Charisma at 22 Or are racial boosts considered one and the same with your ability score?
The maximum is 20 including racial bonus's unless the book specifically states that your maximum stat increases.
PHB Page 15. Some of these abilities allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can't increase an ability score above 20.
Some classes have abilities which state that your maximum ability score increases to a new value.
PHB Page 49 Primal Champion: At 20th level, you embody the power of the wilds. Your Strength and Constitution scores increase by 4. Your maximum for those scores is now 24.
Racial bonuses are considered part of your ability score, which can be inferred from this line on PHB page 173 as it does not specifically say the racial bonus increases your score above 20:
Adventurers can have scores as high as 20, and monsters and divine beings can have scores as high as 30.
You are an adventurer as a Half Elf Bard, the book clearly states that an Adventurer cannot have an ability higher than 20. Unless your class gives you a trait that increases the scores above 20 (like the Barbarian's level 20 feature on page 49 of the PHB) you cannot have an Ability score above 20.
Page 11 has this sentence:
Your race also increases one or more of your ability scores.
Page 15 talks about increasing the scores as you level up not going over 20, which I wont quote here since your question indicates you understand that part of the rules, but I list the page for easy reference.
But it does not say it increases your maximum or the cap. Since the Rules should be followed to the letter if you are playing a game with RAW, no mention of increasing the maximum, means you can't go above 20. Bard's do not get a bonus to any skills like the Barbarian so for your specific class choice you are stuck at a 20 max CHA score.