"remaining connection slots are reserved" during a traffic spike

• Under a traffic spike, requests fail with "FATAL: sorry, too many clients already".
• The database CPU is not saturated; it is mostly idle.

SELECT count(*), state FROM pg_stat_activity GROUP BY state;
  active:               12
  idle:                  4
  idle in transaction:  84
  ------------------------
  total:               100   (max_connections = 100)