The following scenario can happen, and is actually very close to reality: 1) Being vaccinated increases your chance of being infected by covid by16% 2) If you are 60 or older than 60 years, being vaccinated halves your chance of being infected by covid. 3) If you are younger than 60 years, being vaccinated halves your chance of being infected by covid.
In other words, being vaccinated increases your chance of infection, while getting vaccinated decreases your chance of infection.
This is just Simpson's Paradox. Another one example (politics warning) is that Democrats had higher opposition to the 1960s Civil Rights bills but this was due to Democrats having more Southern members. Northern Democrats supported Civil Rights more than Northern Republicans and Southern Democrats likewise supported Civil Rights more than Southern Republicans.
Similarly, I'd wager that the 60+ crowd is overrepresented among the vaccinated so P(infected|vaccinated) is higher because P(infected|old) and P(old|vaccinated) are high.