Or is that too much even with proper diet and rest?

I lift+run on the same day and I'm able to recover. Lifting twice I'd say as long as theyre different muscle groups would be ok

To many factors in play here to narrow it down to a simple yes or no answer

If you take appropriate rest in between training the same muscles you should be fine

On board with this. I push so hard that theres no way I could do chest and later arms.

When I was natural, I was lifting, boxing amateur and doing BJJ all at the same time.

I was able to train 40ish hours a week for about 6 months at peak levels of performance before injuring myself and basically sidelining for three months and having to rehabilitate for another three when the injury continued to persist until almost the point of loss of use of a limb.

Funny, I cannot link the injury to anything (complete reversal of spinal curvature causing it to press against my brachial plexus). Just woke up one day and my arm was completely numb.

Short answer is: yes, for limited time depending on what the activites are. Lifting + running, is fine. Lifting + additionally tasking endurance activity for extended periods of time is not.
Long answer is: Depends on how long you are running it, whether or not you give your body adequate periodic breaks and whether or not you are fortunate.