One can easily train every day and make progress. The question is, how long will you be applying those demands? There is a difference between short- and long-term application, which has a bearing on the dose of the short- and long-term demands. What I can tolerate during a two-week blitz is much different than what I tolerate for 3 months preceding that blitz. The effect is also more profound during that time, but must be short-lived in order to avoid overtraining. Elementary theoretical/practical application of the General Adaptation Syndrome. Of course, this does not address your goals, needs, or reasoning for wanting to do so (viz., why would you want to or need to train for consecutive days?).
I started lifting about 5 months ago, and I started out as a 6'2" 135-pound weed. For the first two months of my training, I trained full-body every day, lifted big. ate bit, and slept big, and as a result gained 45 pounds of solid muscle in those two months alone, not to mention 3 inches on to my arms (which are now 16.75 inches at 16 years old). I still train every day*, many do. Jones was wrong, anaerobic system is, same as aerobic >>made<< for very frequent use. Those two are different, but not opposite.
I believe the less effort you put into your sets, the more often you can (should) work out. If you are stopping two or three reps short of failure, work out every day. If you are working to failure, tossing in some negatives after that, and puking your guts up, you might want to take a few days off between workouts.
Compare and contrast a Pavel Tsatsouline 3x per day workout to a Mike Mentzer 1x per 10 days workout. Both can probably be effective, in different ways, with different approaches.
/Deathspell Omega
* - I now train on a push-pull split, with legs being done on pull day.