Doctor Who aired episodes out of order in 1988 and created a ridiculous plot hole. Modern Doctor Who Viewers are used to the idea that there is a loose narrative flow running through the series, which typically climaxes at the end of the season. Of course, there is some debate about whether Doctor Who Season 14 was broadcast in order, but other than that everything seems to flow very well.
Classic Doctor Who it was a little different. The stories spanned several episodes, but tended to be self-contained. There were times when continuity was important, usually involving important events in a companion's life, but otherwise the stories were all designed as new starting points. This all began to change in the era of Sylvester McCoy's Seventh Doctor, which foreshadows the modern era Who in several ways. Unfortunately, this caused a subtle plot hole that remains quite entertaining to this day.
Ace's jacket has the wrong badge in Doctor Who season 25
The questions flow from Sophie Aldred's book Doctor Who fellow Ace, a character who was something of a pioneer. Introduced as a waitress working on the planet Iceworld, Ace was a far cry from the “damsel in distress” trope seen in some of the older companions. She could defend herself against the most terrifying enemies, throwing Nitro-Nine grenades at her enemies and even defeating a Dalek with a cosmically charged baseball bet.
Ace wore a (then) stylish jacket with multiple brooches and was often seen wearing a cool backpack. Unfortunately, as ridiculous as it may seem, these costume details caused a subtle continuity issue. Ace was shown wearing a new pin on her jacket in "The Happiness Patrol" and "Silver Nemesis" - one she would not acquire until the later story "The Greatest Show in the Galaxy". Even stranger, she looks for her backpack in "Greatest Show", even though it exploded in "Silver Nemesis".
Doctor Who's plot holes were caused by the 25th anniversary
It's not hard to guess that the episodes were shown out of order. The problem was in "Silver Nemesis", which was designed to be Doctor Who25th anniversary episode. Unfortunately, season 25 was delayed by the Olympics, which meant the broadcast order had to be changed. This normally wouldn't cause any problems - classic stories tend to just have looser continuity - but unfortunately, season 25 was different. There were subtle narrative nods throughout, right down to the details of Ace's costume.
There isn't really a solution to this, not even if you try to take advantage of the time. The best approach is to simply observe Doctor Who the 25th season in the order it was supposed to be broadcast, rather than the jumbled order it actually happened. That means checking out “The Greatest Show In The Galaxy”, then “The Happiness Patrol” and only then watching “Silver Nemesis”. Seen this way, the subtle continuity issues are completely erased.