Connection at risk nearly always refers to the amount of time between the time when a flight arrives and the departure time of the next flight for the traveler (on the same day).
Say the initial flight departs late, the traveler could be in danger of missing his connecting flight.
However, that message also can display if the flights are pre-scheduled to have a connection time of less than what the airline declares as a legal connection (different per airport and per airline). Most airlines require at least 30-45 minutes between flights. Some smaller airlines and/or smaller airports (where you don't have to walk far), do schedule flights with connections as close to 15-20 minutes apart. Those flights can trigger a connection at risk flag although having legal connection times.
If in doubt, it's always a good idea to check with your airline or booking agent to make sure the fight times have a legal connection.
Updated: Aug 26, 2016