It was David Tennant's first show as The Doctor, the Christmas Invasion. He spent most of the episode sleeping off the effects of a regeneration, while London was under threat from a big space ship full of bad guys threatening Planet Earth. The Prime Minister was Harriet Jones, who had previously met the Christopher Eccleston version of The Doctor (Doctor #9 or #10 depending on how you count it). Finally, about 10 minutes before the end of the episode, the Doctor wakes up from his nap, he's still in his night gown, and finds himself on-board the alien space ship. His companions, including Harriet Jones, had been doing their best to face down the bad guys on their own but weren't doing a good job of it. As soon as the Doctor wakes up and emerges from the TARDIS, he and the chief bad guy get into a sword duel which The Doctor wins (killing the chief bad guy).
Having won this honor fight, the alien ship has to leave.
After landing the crew back on the ground in London, the Doctor, his companions, and Harriet Jones and her assistant are gleefully watching the bad guys space ship fly away.
Then a call comes - it's Torchwood, an organization commissioned by The Queen to counter the danger The Doctor presents. Torchwood says they're ready, and Harriet Jones thinks for a minute and then gives the order to fire at will. Suddenly several energy beams shoot from around London, aiming at the baddies space ship, and it blows up.
At which point the Doctor gets incensed. He says he should have warned the bad guys - "Watch Out, because the Monsters are coming". The Human Monsters, that is, the ones who'll kill and rape and pillage their way across the cosmos. That sets up a confrontation between The Doctor and Harriet Jones - during which the Doctor boasts he can take down her whole administration with just a couple words.
And then he walks over to her assistant and whispers "Don't you think she looks tired" and walks away. A couple seasons later the next time we saw Harriet Jones, she was a former Prime Minister, and Harold Saxon (The Master in disguise) was the new Prime Minister. But that's another story for another day.
The point of all this is --- that's what the Donald Trump propaganda machine is trying to do to Hillary Clinton.
Yes, okay, today we learned that Mrs. Clinton currently has a case of pneumonia. People get sick don't they? Is it that rare for someone to become sick when they're pushing themselves beyond belief? According to what I've read in the news reports today, the facts of Mrs. Clinton's health is very good - she's active, healthy, has a good diet, rarely smokes or drinks alcohol, etc.
But that isn't gonna stop the Trump people from making A Thing -- concocting false stories about Clinton's health, just as The Doctor did to Harriet Jones. Face it, in that moment the Doctor was not the Hero but a revenge-taker. He concocted a false story to take down the Prime Minister.
Trump and his ilk do not have any pretense for being kind thought-full people who look to create the best in the world. Instead they have no compunctions concocting fake stories to aim for political gains. The stories they'll be spreading against Mrs. Clinton won't have a shred of truth, but that doesn't matter because it's all about appearances. Apparently they've been raising doubt over her health for several weeks now, and they will surely leverage this all they can.