I found a couple of articles written by a Dr. Stuart McGill that walk about the "hydrophilic" nature of the spine.
The first seems to agree with what your physiotherapist said but it a slightly different language. Not that they will burst like balloons but that the are full like a water balloon.
The disks in between each of our vertebrae are packed with very concentrated protein chains that love water. In scientific terms, this means they are “hydrophilic.” When we lie horizontally, the discs fill with fluid and gently push the vertebrae away from one another, lengthening the spine. The reason, our backs are often stiff in the morning is that the discs are so full of fluid, like water balloons ready to burst. 
The same article goes on to say that this water seeps out after an hour or two, and the spine returns to its normal length.
In this article he mentions:
Eliminating spine flexion, particularly in the morning when the
discs are swollen from the osmotic superhydration of the disc that occurs with bedrest, has been proven very effective with this type of patient
From this I can gather that there is likely a stiffness that remains in the spine after waking. They suggest to clients working with back pain to not workout in the morning. Also, they specifically mention avoiding spine flexion -- keeping a neutral spine in workouts can help here.
From a personal perspective: I, as well as a dozen other friends I know, have all had morning power-lifting workouts. From the time I wake up, eat something, get ready, head to the gym, warm-up, and actual start working out it is past an hour.
An import aspect is going to be your warm-up. Your warm-up is necessary if you're working out first thing in the morning as your body hasn't had a whole day to move. Get your body stretched and warmed up before you start lifting, keep a neutral spine and always focus on technique, and I think you should be plenty alright.