A hung parliament is a realistic possibility even today. Even in an age where there are deep divisions in political parties and beliefs, there are ways for government parties to be hung on a ruling. If enough people simply have two separate beliefs in completely equal numbers, then this is all that is needed to have such a quandry.
A hung parliament occurs in a two-party system where neither party holds a majority of the seats. When translated to the American government it would mean that neither the Republicans nor the Democrats hold the majority. So each may have 45% and the other 10% of seats are held by minority parties. It generally doesn't happen in the United Kingdom and it has been uncommon there since WWII. With parties springing up in the United States and the G.O.P. falling out of favor, it is far more likely in the United States within the next ten years. Of course, then it wouldn't be called a hung parliament because we don't use that structure.
Though we haven't seen a hung parliament in quite some time, it is still a realistic possiblility. If the United States government can shut down over a debt ceiling extension, then surely a flawed parliamentary system can still be hung if things go the wrong way, or if there is no give
A hung parliament is not possible today because members are too strongly aligned with party interests. Independent thinking and doing what is right regardless of political consequences is on the decline. The populace is also too politicized to support members and candidates who do not toe the partyline. In essence, everyone is too in it for themselves.
No, I do not think that this is a realistic possibility here in today's world, and I do not think that having a parliament is a good way of governing. The best form of government is like the United States, where you have a balance of powers spread out even.