Granted, there is a correlation between excessive REM sleep and not getting enough deep sleep, as well as excessive REM and depression. However, this usually includes individuals who are also getting excessive sleep in general (over ten hours). The quality of emotions during the day may play a role too as intense and negative emotions are linked with higher REM cycles for depressed people. But for people without depression, there is not a link between intensity of daytime emotion and the amount of REM sleep.
This trend of negative emotions also can be seen in dream content. For example, for most people REM dreams start out with negative content in the first part of the night, followed by dreams with more positive outcomes in the morning. But for the depressed person, these later dreams are also mostly negative, possibly reinforcing depressed feelings during the day.
If you are having feelings of depression, anxiety or emotional overload often, you may want to consult your medical practitioner about getting a psychiatric evaluation. Many prescribed antidepressants reduce REM sleep as a side effect, and some psychiatrists believe that this reduction of REM may aid in the process of overcoming depressed thinking, especially if many of your dreams are usually negative or nightmarish.
For those who do not report any feelings of depression or anxiety, a good self test would be to reduce your sleep by waking up a little earlier, therefore cutting out your last REM cycle. After a period of a week, see if you notice any differences in tiredness or mood. Just be careful not to restrict your sleep too much - making yourself even more tired wouldn't be good!