Hello Ultra Sceptile,

TL;DR: Yes, though not necessarily.

For the dragonborns met during the Chroma Conclave arc of Campaign One, not all dragonborns have an innate source of magic. Based on this observation, having dragonborn in your character's bloodline would not guarantee an innate well of magic such as would having a black dragon in the character's ancestry.

Ultimately, this is up to you and your DM. You could have a dragonborn in your character's family history whose line has an innate source of magic. This could be due to a long forgotten connection with a god (prime, lesser, betrayer [like Tiamat], etc) or patron (either of which could make for exciting character development). If it were long forgotten, perhaps your character operates on the assumption that it is due to the known dragonborn ancestor without realizing what an innate well of magic really says about their ancestry/significance. The world's your oyster.

Hope this helps! Best of luck on your adventure!

• This came up on my feed, but much too late. Perhaps this can help those who stumble across this.

I’ll be honest I completely forgot about this post, but thank you for giving me some pointers, as well as anyone else that sees this

That is kind of you, @The Ultra Sceptile —I felt bad when I saw the date... Anyhow, I hope your character's development has been fun and engaging in your game!