Yes, there were many times when there was a serious possibility that the Stuarts would be restored.

James II tried to get his throne back with French help by invading Ireland in 1689 but was defeated at the Battle of the Boyne in 1690. In 1692 the French assembled an armada of 44 ships to invade England and restore James to the throne but this was defeated by the Royal Navy at the Battle of La Hogue.

1708 also witnessed an unsuccessful French invasion attempt involving James II's son, also called James. 1715 saw James travel to Scotland and raise a rebellion to try and win the throne. However the forces he mustered were defeated and he had to flee back to France. His son (James II's grandson) Charles managed to land in Scotland and this came closer to success: he invaded England and reached as far south as Derby in the Midlands but instead of taking London his council got cold feet and they retreated back to Scotland. They were eventually defeated.

It later emerged that there was a Hanoverian spy in Charles' camp who helped persuade Charles to turn back from Derby by claiming there was an additional Hanoverian army waiting for them if they marched on London (this was false). This seems to me to be the closest the Stuarts came to a new restoration.

The Whig governments (who were in power from 1715 onwards) suspended habeas corpus at various times during the eighteenth century when there was a threat of Jacobitism. Jacobitism was widespread throughout the British Isles, along with the kindred doctrine of the divine right of kings. Due to it being treason to express Jacobite sentiments Jacobitism had to be expressed covertly in code (for example if you were to say of a gentleman that he is an 'honest man' this was an inexplicit way of saying he was a Jacobite). However due to the unrepresentative nature of Parliament at this time Jacobitism did not gain power commensurate with its popularity. The Tories were a largely Jacobite party and they would have won every general election between 1715 and 1747 had the number of seats obtained corresponded to the number of votes cast.

As late as 1828 a daughter of George III, Princess Augusta, remarked: “I was ashamed to hear myself called Princess Augusta, and never could persuade myself that I was so, as long as any of the Stuart family were alive; but after the death of Cardinal York [the last direct Stuart claimant, in 1807], I felt myself to be really Princess Augusta”.

Source: Paul Kleber Monod, Jacobitism and the English People. 1688-1788 (Cambridge University Press, 1993).