I did it. I applied to graduate school pregnant with my first (despite everyone telling me that I'd never get in, applying as an obviously pregnant woman to a very competitive PhD program.) I did get in, to my first choice program, and proceeded to have several more children along the way. I think it depends on your outlook. In my culture, getting married young and having children was a given. Everything else had to be figured out. If you're determined enough, you can do anything. Yes, there were some graduate school experiences that I missed out on along the way. Yes, there were probably times I was passed over for opportunities by old-fashioned faculty who assumed that a mom of young children wasn't "committed" enough (because of course, having a 4.0 GPA, and a full load of research and externship obligations, while lugging a breast pump to graduate school means you are somehow a dilettante.). Yes, there were probably things I missed out on with my kids. Fortunately, I'm married to someone who is an amazing father, and we both realized early on - forget traditional gender roles, forget how it seems to others - the key thing is that the child has A parent with them at a critical time. Doesn't matter if dad is the one baking the cupcakes for the class party, what matters is that cupcakes get baked for the class party! It really all depends on your expectations and what you are willing to forgo. For me, it was pretty much - a social life, a decent wardrobe, and just time for myself. While in grad school, those things were on hold. Nice thing about grad school is, it ends.
Internship was the hardest year, it meant I had to recruit a ton of support, like full-time cleaning help and help from the family. It still was a really challenging year - successful, I learned a lot and made lifelong collegial relationships and friends, but challenging. I could not have done internship pregnant or with a newborn. When I was on internship, my youngest child at the time was 3. 3 year olds need a lot of care, but they don't wake up every two hours at night and they do spend a portion of the day in preschool.
I'm now working for a private practice as a clinician while I wait for the glacially slow NJ Board to fully license me, which should be within the next two months. Working for a private practice as a mom is totally do-able and manageable - I am able to set my own hours (more or less, given patient availability). Once I'm fully licensed, my income is set to double, especially since the practice I'm working for is being very generous about allowing me to bring my patients with me to private practice (technically, no one "owns" the patient, but they could have made it much more difficult).
You really have to be realistic about what you can and can't do without, and make decisions based on that. I think a supportive partner and an extended circle of family or friends is key.