As a native speaker who is 100% confident of my correct usage but equally unsure as the BBC when it comes to explaining...
- think of: recall (memory) or conception (new idea)
- think about: analyse
"Thinking of buying a new car" sounds a lot more obtuse/non-committal than "thinking about buying a new car." The former sounds like you're toying with the idea, whereas the latter sounds like more serious planning.
Similarly, a card saying "thinking of you" sounds more like sympathising, while "thinking about you" could potentially imply willingness to act/help.
You can think about (analyse) something deeply. In general you can't really think of (remember/conceive) something deeply.
Whenever there's a scenario where both "about" and "of" could work (such as with the cars and cards examples), listeners are unlikely to be attentive or pedantic enough to really register the difference. It may be subconscious at most.
I thought of an answer to your question. Then I though about the answer some more before posting some edits.
You cannot swap the of and about in this sentence.
Examples of incorrect use
It all depends on what you actually mean to say. It most cases, the following are wrong because they don't convey the intended meaning:
I always think about her.
Probably wrong since you are likely remembering rather than analysing.
I think about being asleep at home when I am at school.
Probably wrong since you are likely not analysing. You're probably also not thinking, but rather dreaming/yearning/longing ("dream of being/yearn to be/long to be").
I am thinking of buying a car.
Probably wrong since you are likely analysing the pros and cons.