I did it with two kids (small cohort, funded, university PsyD program). A few quick things:
You won't get home by 5pm. I got home at around 8pm most weeknights. A lot of the time, I sent my kids to school in the morning and didn't see them until they woke up the next day. I worked on weekends. You have to. It caused a lot of stress and I felt like I missed a lot of my kids' lives during this time. It's VERY hard.
In order to do this as a parent, you have to be
extraordinarily organized and time efficient. You NEVER procrastinate because, as a parent, there is no guarantee of available time in the future. Wait until the day before your paper is due to work on it and I guarantee you your kids will have a snow day and your partner will be working all day. You do everything as early as possible. You will have a lot less time than your childless cohort-mates. You have to be in beast mode the entire time or you will fall behind. You also have to effectively manage your stress while you are parenting.
@Sanman said earlier that no one will care about your schedule and personal responsibilities and I also found that to be true. The program is not going to adjust around you, you need to adjust around your program. Same for kids, right? Your academic schedule will be unpredictable and you won't have flexibility. Having children is also unpredictable, also without a lot of flexibility. Put those two together and it's extremely stressful.
In addition to scheduled classes, practica, etc., my program had a number of "required" events that were often scheduled with short notice. E.g., "All second year students must attend this two-hour training next week" and it would be on a day I hadn't even planned to be at the university, offered only once, and an absolute requirement. It's very hard to plan ahead, you're essentially on call for your program.
You will need A LOT of flexible, reliable childcare. Childcare is expensive and unless your partner is a full-time stay-at-home parent, you'll probably need to tack on childcare costs to the overall cost of your education. I was fully funded and took out student loans to cover additional childcare.
This sustains for a minimum of 5 years. It's hard. I gave it a lot of thought before starting the program and there were still a lot of challenges I didn't predict.
I don't regret it. I am passionate about the work and truly enjoyed life as a grad student (minus the parenting stressors), which motivated me through the process. I loved research, I loved writing my dissertation, I loved the lectures, I loved learning and challenging myself. I do pretty well in high pressure settings. It helped that those things were reinforcing to me (they are not to everyone). Now that I'm done, I have a lot of flexibility and spend a ton of time with my family. I love what I do and I love my current life balance, but it was a long road to get here. You just need to think ahead and make sure you are prepared for the realities of this path.