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.
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.