Particularly after the Industrial Revolution in England began, it was increasingly possible to be rich in a lower class. There isn't a good example of this in Persuasion because the central characters are for the main part in higher social classes either because of inherited family fortunes or because of newly gained fortunes, such as Captain Wentworth's wealth. However another one of Austen's novels illustrates the concept of being in a lower class but being wealthy. In Pride and Prejudice, it is speculated that Mr. and Mrs. Gardiner would not be worth knowing by anyone in a higher social class because they probably gained their income through the lowly work of being merchants. Merchants and other tradesmen were lower class people but some were newly wealthy because of the Industrial Revolution and the vast number of imports from the various parts of the British Empire.
It was equally possible to be in a higher class and have no money--but not for long. Anne's father, the Baron, illustrates this very well. The reason they have to leave the family estate and move to cheaper lodgings in town is because the Baron couldn't control his spending and had squandered his fortune--and his daughter's fortunes--away on keeping up a lifestyle he thought appropriate for a Baron, which means it was an expensive lifestyle. Austen's other novel, Pride and Prejudice, also illustrates this point very well. Colonel Fitzwilliam is the younger son of an earl and therefore not the future recipient of his father's title and fortune. He has and will have no independent wealth of his own. Yet, as the son of an earl and the nephew of Lady Catherine de Bourgh, he is in a higher social class.