Suppose $N$ players play a game, where each player's action space is $[0,1]$.
Each player has an identical continuous utility function $u:[0,1]\times [0,1]^{N-1}\rightarrow\mathbb{R}$, where the first argument is their own action, and the other arguments are the actions of other players, with $u$ being invariant under re-orderings of the other players.
Suppose that the game has a unique pure, symmetric, Nash equilibrium with each player taking action $a\in (0,1)$.
Now suppose that the game's rules are changed, and players can now only choose actions in the interval $[b,1]$, where $b\in (0,a)$. By revealed preference, each player taking action $a$ is still a Nash equilibrium (if they did not have a profitable deviation before, they do not now).
Is it possible (for some $u$, hopefully not too weird!) that each player taking action $b$ is now an additional Nash equilibrium, even though it was not before? Is it possible that the $b$ equilibrium survives some refinement (e.g. trembling hand) that the $a$ equilibrium does not?
Now this has been answered, I'll briefly note the particular circumstance I had in mind. The $N$ players are countries, choosing the level of their corporation tax rate. $a$ is the moderate level of corporation taxes (most) countries have set in the absence of international agreement. $b$ is the new minimum level now internationally agreed. Whether the $u$ in the answer looks anything like countries' pay-offs in setting their corporation tax rate is another question!