Having children while at graduate school
Anecdotes are not usually the right kind of answer on stackexchange, but you have specifically asked for them, so here goes...

My experience

I had two children in graduate school.  The first was born during my second year, and the second was born during my fourth year.  Most of my fellow students thought I was crazy.  I graduated (obtaining both a MS and PhD) in five years, with no debt and certainly no regrets.  Surprisingly, I found that graduate school allowed me to be more involved in my children's life than if I'd had a regular job.

Advantages

Some important and perhaps unusual factors worked in my favor.  First, and most importantly, my wife had already decided that after we had children she wanted to be a full-time homemaker.  Her support allowed me to focus on my studies, although I did change hundreds of diapers, stay up a lot of nights, and after the birth of the first I didn't get any research done for about three months (I did continue passing my courses).

The second important advantage I had was generous funding from the US DOE's computational science graduate fellowship (CSGF).  It is (or at least was) the highest-paid government fellowship in the US, at about $33K/year.  That was still officially below the poverty line for a family of four where I lived, and half my income went to rent, but by careful budgeting we had no problem getting by on only that income.  Could I have done it with less income?  Yes, though I would possibly have needed student loans, and the added load of being a teaching assistant would have been a challenge.

Finally, I found that I could get a lot of high-quality research done by focusing intensely for a few hours per day.  For me, that was (and is) a more productive strategy than trying to focus on research 24/7.  This made it possible to complete my thesis and care for my family simultaneously.  I was home by 5-6 p.m. almost every evening and spent most of my weekend time with my wife and children.  I did often do research for an hour or two after my kids were in bed.  I know that many grad students tend to spend a lot more time each day on research, which might make things more difficult.

Assessment

Having children during graduate school was a good decision for me.  The time I was able to spend with them as babies and then toddlers is precious to me.  After I graduated and became an assistant professor, I found that the time I could spend with my children was much more limited, due to the added demands of teaching, advising, service, and administration.  Indeed, with my third child (born 2 years after I got a permanent position) I changed a lot fewer diapers and really spent a lot less time with her early on.  I've reached a better balance since then, but it took years.

In short: with the right setup, having kids in grad school can work out very well.