Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Situational Awareness
Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude
Flight Crew Took Evasive Action
Aircraft; Human Factors
The aircraft was at 5;000 feet on an a downwind operating under IFR conditions with the final Savannah Approach Controller.  A weather cell was in front of us approximately 15 miles away.  We advised we would need right deviations.The Controller had both VFR and IFR aircraft under his control.  I observed an aircraft in our vicinity on our TCAS with a sporty vertical closure rate.  The aircraft was identified as a small aircraft.  The pilot was informed to stop the climb at 4;500 feet (this tells me the pilot was cleared to a higher altitude which would have conflicted with our flight).Though the pilot flying was the First Officer; as the closure rate continued and the colors/threat level changed; I verbalized and disconnected the aircraft autopilot and added a small amount of thrust as we received a red TCAS and a subsequent climb command.  I informed the FO to start climbing (I had a little pressure on the yoke to begin the climb).  A smooth climb initiated respecting the RA information.  Once the conflict was resolved; we leveled off (approximately 6;500 feet) and maintained our course heading (270 degrees). Sometime during our climb and a lower workload; I communicated our TCAS RA to ATC.  We were then given a descent to 4;000 feet.  According to the First Officer; the RA depicted our vertical separation of -200 feet.  With that information; in my opinion the event qualified as a near miss.  When workload permitted; I simply advised our event was a near miss.The Controller responded the conflict aircraft was supposed to stop at 4;500 feet.  The conflict aircraft then communicating and over-talking on the frequency.  I simply noted we are all fine----we would like to clear up the frequency.The flight landed without incident.  During deplaning; I listened for passenger comments to determine if anyone was aware of the alert.  I could not determine any customer who asked about the event during the approach phase.  Later in the day; I asked the Flight Attendants if they noticed or heard anything unusual going into SAV.  The lead FA was not aware of anything.  The rear FA noted she felt the aircraft climb for a short period.  Their feedback is consistent with the physics felt during the climb.Dispatch was notified.  No delays were encountered for our return flight to NY.Cause: Controller workload and frequency congestion.  The conflict aircraft also operated under VFR conditions when it was extremely hazy with weather surrounding the area of conflict.  It is easy for me to opine the aircraft should have operated under IFR; however; that is a biased opinion.Suggestions: Clearly our manuals provide the guidance and years of experience provide intuitive actions.  Regrettably; we have not reinforced the 'positive aircraft control' concept in our training evolutions; even after documented events with similar recommendations.
CRJ-900 Captain reported climbing in response to an RA with an aircraft 200 feet below them on arrival into SAV.