I'll bet this is a really common problem, S., so I'm glad you are asking. It's important to know that our children are unique beings, separate from us and from each other. Just as a surgeon can't work effectively if he's too identified with his patients' suffering, we can't do our most effective parenting when too closely identified with our kids' emotional states.
It's excellent, I think, to be sensitive to their feelings, which will help keep communication open and compassionate. But when the connection is too tight, there's little breathing room left for either mother or child. That's just not healthy. And (sigh) there can be a mile between knowing that in your head and really getting it in your heart.
If your children are still very young, some of your identification is normal, and reinforced by hormones – one of the ways mama creatures are prompted by nature to care for their endlessly demanding little ones. The intensity will probably mellow through the coming years.
If that doesn't occur, you might do well to get involved with interests of your own, a reading, exercise or crafts group, volunteer work if you're a stay-at-home mom. You could even do something with your children when they're old enough, broadening the social network for all of you.
If the intensity of your connection is just too draining now, you might want to talk to your doctor about it. Counseling might be in order, or possibly even medication depending on any other emotional or behavioral symptoms you might be having. Some women are simply sideswiped by the impact of motherhood. Depression is not merely a rumor.
But chances are, you're just a normal, caring mom. I see the point in what your mother told you, but if you start to inquire into the nature of your happiness, you can gradually learn that you can be happy, well-balanced, and productive even if one of your children isn't.
I know a woman who's son is profoundly autistic, and frequently shows signs of distress. She's a cool lady, though, and has learned to detach her need for happiness from her son. She's a terrific and caring mom, sacrifices a lot for his well-being, as well as that of an older daughter, but she simply doesn't get dragged down by the fact that he'll never function quite normally in the world.
A quote comes to mind from a wonderful poem by Mary Oliver:
Tell me, what is it you plan to do – With your one wild and precious life?