Is there a Card Cap?

Solution 1:

As there is not offical statement on the card limits, at worst there would be some programming related limit.

At the very worst this number of cards would be stored in and signed 8-bit number variable, putting the maximum number of cards at 127.

Most likely the number is at least 8-bit unsigned integer (because you can't have negative amount of cards), putting the maximum at 255.

16-bit integer is also possible, when the maximum would be 32,767 or 65,535 (depending on wether the number is signed or not). But then Blizzard would need double the space to store people's collections.

In short, unless you think you have literaly hundreds of copies of a given card, you shouldn't have to worry.

(Assuming Blizzard isn't doing something incredibly fancy trickery, but that would be impossible to speculate any limits on.)

Solution 2:

No. I've seen plenty of streamers getting 9+ cards and they've kept opening cards.

Blizzard hasn't mentioned anything about a card cap either, so you're safe.