Yes this pain could be because of your bad forms during core training.
Assuming you get this back pain only at night, may be between 2 AM to 4 AM. Sleeping on your stomach will make the pain worst. and sleeping on your back will be better than sleeping on side. These problems usually starts around age of 40 and/or if you have a belly. If you had more stress at your work, you will have more pain at night.
Usually this happens because of bad form of sitting/walking at work or during workout, after gaining weight body posture and form can go bad. Things like putting pillows around you to make sure you sleep on your back whole night may help. Or when you sleep on your side, put a pillow between your legs at knees could help.
But the actual problem could be your bad form in your day to day activities or during your core training, if you have weaker abs (specially lower abs) then you might be compensating it with your paraspinal muscle. The same way sometimes pregnant ladies get lower back pain. Technically (Medically) speaking The deep muscle Adbominus transversalis and Erector Spinae and Superficial muscles like Rectus abdominis and Pectoralis major and even Serratus Anterior (and I guess even Iliacus and Psoas major) all these muscles work together for spinal movements, so depending on where exactly you feel pain and how much deep, Chiropractor or Physiotherapist can help.
I don't want to scare you but if you continue doing core training wrong way, or if you don't fix your posture in your day to day activities, you can get more serious problems like herniated disk in a long run.
Ps. I am just a software engineer, with an interest in body building and human anatomy.