Is the story of Pride and Prejudice believable?
Certainly the novel is realistic in portraying typical characters often seen in specific households, and in bringing up topics that were very talked about in Regency England such as marriage (and the need for it), entailements, social ranking and shunning, the quest for love, and the position of women in society.
In these aspects the story is quite believable. What you may be refering to is if its comedic and satirical tone makes it less realistic and harder to take seriously. The answer to that is that Austen had precisely that purpose in mind: To entertain, as well as inform.
Lastly, this genre is called "Comedy of Manners" whose specific purpose is to satirize and exaggerate the behaviors of the likes of Mrs Bennet, Lady Catherine, and Mr. Collins.
Now, did the situations occur back in the day? Yes! Entailements, husband-chasing, and gossip were all staples of that society and it is a very realistic scenario.