Along with the distinctive cry, there are a few other signs that your baby might be too tired to sleep. Overtired babies tend to fall asleep for short naps rather than longer stretches of sleep, start falling asleep at random points throughout the day, and (most frustratingly) will fight sleep when you put them down in their crib.
You might even find your baby waking after they've already fallen asleep, despite the fact that they clearly need to catch some Zs!