Yes it is possible to sometimes have the cold and flu virus at the same time and I've had the flu and virus at the same time myself.
Although it's rare for me or anyone to get the cold and flu virus at the same time it is possible.
The flu and common cold virus are caused by different families of viruses so that is one reason that makes it possible for both types of viruses to enter your body and attack your body at the same time.
If you have a cold then you can later get the flu virus while still having the cold virus and having both the cold and flu virus in your body at the same time will make you feel even more miserable than you would with just the cold or flu virus alone.