Well, making yourself immobile is a no-brainer as far as being unproductive.
If you workout the entire body too hard you may find yourself not just very tired but exhausted on a deep level. Your concentration and overall feeling of well-being could be affected, and you could be very sleepy or make your immune system more vulnerable. In short, don't overtrain.
But if you just give your whole body a lighter level of stimulation, it can be very good for you, especially if you include stretching to maintain your flexibility.
Just don't burn-out. If you feel like pushing very hard, restrict the body parts. Like if you decide to go psycho in your conditioning by running for miles and miles and also doing a ton of push-ups and dips, leave your back and pulling muscles alone. There is such a thing as overstimulation, and your immune system definitely can be weakened if you do too much. At that point, cold or flu germs already in your body that your body has been resisting may suddenly get the upper hand, and new ones you would have fought off this time you can't.
I say this as someone who was an extremely committed martial artist for a number of years and overtrained far too often. Sometimes I would come down with a cold practically DURING a particularly brutal work-out, from a point of apparently perfect -- even outrageously great -- health before the work-out began. There's a lot to be said for having more unstoppable determination, but like it or not, the body is more limited than the willpower. Even if it carries you through basically the physical assault on yourself that an extreme work-out can be, that doesn't mean you won't have worked it to a state of great vulnerability after.
At that point, it's probably really not worth it. It's overall better to give a little less than your all in a work-out, and hold back a little bit extra. That reserve or little bit extra could actually be your freedom from injury or overall systemic health.