They absolutely do. Since it's world cup time I'll use a contrived soccer example.
A penalty kick where the kicker can kick left or kick right. The goalie has to jump one direction, if they jump the wrong direction a goal is scored. Both people know that this kicker is great at kicking to the left side of the goal but rather "meh" at kicking to the right, so if the kicker kicks to the left and the goalie jumps left, there's still a 20% chance of scoring, but if the kicker kicks to the right and the goalie jumps right, there's only a 5% chance of scoring.
There is a Nash equilibrium for the kicker, and it can't be "always kick left" because then the goalie would "always jump left" which would give the kicker an advantage if it kicked right.
Similarly the Nash equilibrium for poker can't be to always fold a weak hand, because that's leaving money on the table because then the opponents will always fold against a raise, which would mean the player could get easy money by raising with a weak hand.
A penalty kick where the kicker can kick left or kick right. The goalie has to jump one direction, if they jump the wrong direction a goal is scored. Both people know that this kicker is great at kicking to the left side of the goal but rather "meh" at kicking to the right, so if the kicker kicks to the left and the goalie jumps left, there's still a 20% chance of scoring, but if the kicker kicks to the right and the goalie jumps right, there's only a 5% chance of scoring.
There is a Nash equilibrium for the kicker, and it can't be "always kick left" because then the goalie would "always jump left" which would give the kicker an advantage if it kicked right.
Similarly the Nash equilibrium for poker can't be to always fold a weak hand, because that's leaving money on the table because then the opponents will always fold against a raise, which would mean the player could get easy money by raising with a weak hand.