Is there any limit to how many badges you can get in a day?
There doesn't appear to be a limit. Here's what the website says:
Badges are awarded on SpanishDict for achievements big and small. You can earn a badge for posting a good answer, completing a language course, or for making your first flashcard set. You can even earn a badge for following us on Twitter! Check out the badges below, and start collecting them today.
No sé, ¡Lo siento!