To a point, yes. You need to start worrying if it gets so slow that it's about to stop though
Generally, thinking of your heart as a muscle, the fitter it is the less hard it has to work to pump your blood round your body. By the same context the weaker it is the more times it has to pump to get your blood round.
Exercise makes your heart fitter, obviously, so it stands to reason as you run more your heart rate slows and you feel fitter.
However, if you're coming down with something or ill, then your heart rate natually increases, which is why many athletes check their resting heart rate before they get up in the morning and some take a rest day if it's 10 beats or more per minute above what it normally is.
Mine fluctuates because I'm asthmatic so I guage my fitness in other ways and I won't run with a cold, no matter what type of cold, or if I'm generally not feeling 'right' or too tired.