Yes, Gayle, they often do. In a way it is a good sign, something of an indicator of developmental progress. It shows that she is now a thinking child. She is aware of the passage of time and now knows she won’t see you for several hours. She is also letting the world know she is “attached” to you and that you “belong” to her in a special way.
Your handling of this is important. Try to get to her nursery school early enough to allow you to remain a little while, so she can integrate you into her concept of the preschool. Then give her some concrete cues about when you'll be back: “You’ll take your nap and then have some juice and go outside to play. Mommy will be here to take you home right after that.” And make sure you keep your promise. She has to be able to believe you in order to trust you.
Finally, since you say this happens only when you leave her at her preschool, let me suggest you spend a day observing there. Something may be happening that is upsetting her, causing her to resist going there without you. Always look for simple solutions first.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.