Fellow BJJer and weight training enthusiast here. The good news is, it depends. The bad news is, it depends.
Seriously, though, if the basic question is whether or not the human body could perform two training sessions a day, well, sure. The key to this would be how much are you able to tolerate it or, even thrive, during it.
And that's the "bad" news.
Not so much "bad" as it would require you to experiment and be cognizant of your reaction to any of the variables. Be sure you're keeping a log of your training. That way, as you are trying to evaluate your body's response to the training stress, you're clear if you are trending positively or negatively as you change the experiment.
The variables I would watch are:
3. Environmental stress (work, school, social)
4. Intensity of training
Obviously, you would want to ensure you're tops in sleep and nutrition (and tracking that), trying to keep your stress levels low and monitoring the intensity of either of your training sessions. The logs will help you to ascertain what level of training you can handle and empirically define if you're doing well (getting stronger, rolling with better intensity, etc.).
As you define your baseline (e.g. I need this much sleep, this much/kind of food, etc.) you can then calibrate your workouts if any of those variables are different than normal. For example, if you're less rested than normal, if you're nutrition sucks or you haven't recovered from a particularly difficult weight training session, you can vary your workouts so you risk less injury or beat up your immune system and get sick.
I've been doing BJJ for about five years and have been a gym rat for 20. As a busy professional with a huge family, I have to make all the parts fit well in order to maintain this high level of commitment. Good luck.