Amusingly enough failing to pay utilities results in some of the most lenient reactions possible; partially because many localities make it illegal to remove electricity/gas service during "extreme weather months" (because old people freezing to death because they didn't pay a bill is really bad publicity for the utility and the government) and partially because they know they'll get theirs eventually.
They will cut you off when the time comes, but it can sometimes be surprisingly long. And they will put a lien against the property, too.
Having been in this situation myself, no they did not give a lenient time. I got a letter stating that they would shut off my electricity and gas. I had a mere couple of days to sort it out.
In my case it was their fault. I configured automatic payments and they migrated their online system and subsequently dropped my data for payments. Still, they were ruthless about it.
What I've usually seen is they threaten to shut you off with very strong wording, and then continue to threaten you for 3-6 months before they bother to send someone out.
My experience with LG&E and Time Warner in Kentucky circa 2008 was getting a shutoff notice dated the day after the payment due date with 14 days to comply. They did in-fact shut the power off at the state date.
This was a common occurrence that people referred to as 'brown billing' because the shutoff notice had a brown header instead of the green header on your regular utility bill.
They will cut you off when the time comes, but it can sometimes be surprisingly long. And they will put a lien against the property, too.