I can't speak for Britain specifically, but there have been a few instances of this occurring in Australia - which derives its parliamentary conventions directly from Britain.

All that's required to form government is to get a majority of members of the lower house (House of Representatives for us, House of Commons for you) to vote for it. In the first decade of Australia's history as an independent country in 1901, no party ever had a majority. The first majority government in Australia occurred after the election of 1910.

Before that, there were three roughly equal parties in the Parliament. One Prime Minister of the time referred to the situation in the following way:

What a game of cricket you would have if there were three elevens in the field instead of two, and one of those elevens sometimes playing on one side, sometimes on the other, and sometimes for itself.

There were three parties in the Parliament, of roughly equal size. The only way to form a government was for two of those parties to form an alliance, which would result in a majority. And, those alliances changed from time.

There was even a brief period in 1904 when one party ruled alone.

After the election of 1903, the Protectionist party governed with the support of the Labour party. This alliance fell apart in 1904, and the leader of the Free Trade party (the Opposition) declined the Prime Ministership. Therefore, the leader of the Labour Party ended up in government, in the ultimate minority government: not only did Labour have fewer members (22 out of 75) than either of the other two parties, but it also did not have the support of either of the other two parties.

Australia was once led by a party with support from less than one third of the lower house, and Australia's parliamentary conventions come from Britain.

This question has been removed as it is not about history.

Are you asking only about a specific country?