Is it possible for mains power to drop during peak times?
Yes, it is perfectly possible for the mains voltage to fluctuate at peak times.  Electricity companies work very hard to try and predict when peak consumption is going to be and bring extra capacity on-line.

They have to take into account such things as peak TV viewing, when advert breaks happen or popular programs finish (as people get up to turn the kettle on and make a cup of tea - a surprising drain on the grid when everyone that's watching East Enders goes to make a cuppa after it's finished), big football matches, etc.

And yes, there is a natural rhythm to the power consumption during the day/night cycle.  Night time is often billed at a lower rate because there isn't the demand as much at night.  That's not just to be nice, but to try and encourage people to move some activities to nigh time to try and ease the load on the grid.

Evenings are the absolute peak time for household consumption.  People watch TV, have lights on, etc, which they don't do during the day because it's light outside, and they are often out at work.

Business consumption tends to be far more even and regular.  Servers and banks of big computers, which are some of the big normal power consumers, are often running 24/7 - it's only the desktop computers (and not always those) and lights that change the load.  Big industrial machinery is a special case and often has its own arrangement with the grid if massive amounts of power are needed.