You certainly can train seven days a week, but not with the schedule you mentioned. However if you train 3 or 4 main workouts and then add in special workouts (to raise work capacity, work on weak points, or restorative work) as you would in a westside split, then it works just fine. Just make sure the special workouts have a purpose and are short and concise.
For more info check out the articles Louie Simmons wrote (on Elitefts site). He mentions in one article about doing on average 12-18 workouts a week, and that special workout are most important for drug free lifters. Chad Waterbury has also mentioned doing extra workouts as restorative work. As far as main workouts
though, that would be overkill.