Yes, fetal movement CAN be an indication of serious problems. My first baby was very active, and he'd usually wake me up a few times at night due to his constant movement. Then one morning, at 36.5 weeks, I woke up and realized I had slept through the entire night. I got ready for work, and then sat in my driveway debating whether to go to work or to call my doctor. I went back in the house and told my husband that I didn't recall feeling any movement during the night. We called the doctor and she had us come in for monitoring.
During the 30 minutes that I was on the monitors and ultrasound at hospital my baby didn't move a single time. The doctor was even pushing pretty forcefully on my uterus with the ultrasound probe. They gave me orange juice and candy--nothing. So, the doctor decided he needed to come out. They gave me a bit of pitocin to see how the baby would tolerate labor, but his heart rate dropped immediately, before I had even felt my first contraction.
When they got him out they saw that the cord was wrapped around his neck. We had gotten him out just in time. Of course, no one told me the extent of the emergency until several weeks after birth. He was in the NICU for 13 days, but I just thought it was for prematurity and lung development.
So, with each subsequent pregnancy I have been given weekly non-stress tests starting near the 30 wk mark. Sometimes the baby is sleepy and doesn't move much, and I start to worry, but we'd rather be safe than sorry.
Good for you for paying attention to fetal movement! My OBs say that it is rare for the mother to be in tune enough with their baby's movements to realize fetal distress.