It's theoretically possible but would need to be performed carefully and with forethought in order to be reliable. Intoxication with alcohol presents the risk of being sick (if you have little tolerance) which would be a complicating factor, benzodiazepines or another sedative would aid in suppressing any survival instincts potentially preventing the body's instinctive drowning response (which is not pleasant) from being effective.
Unintoxicated, one of the risks of failure here would be the instinctive response of the body which would be to push up against the bottom of the bath and tilt your head back (lifting you above the water), another risk is that any violent movements would cause water displacement, again potentially lowering the water level and making the drowning response (survival instinct) more effective (less height for your head to clear the water-line).
I have read case studies of bathtub drownings where people have been found face-up and face-down, one person lay face down and placed weights on their back to ensure they wouldn't surface. Although it's probably common sense, I would also recommend against using any kind of bubble bath or other products as these will irritate the eyes and lungs presenting further potential complications.
It's a viable method that with some advanced planning may be suitable, but the devil is definitely in the detail. Drowning whilst conscious is extremely painful, I've experienced it myself and can attest to that, therefore careful consideration should be given to any cocktail of alcohol or drugs ingested - quantity does not equal effectiveness (too much alcohol or too many of the wrong medication may be counterproductive).
I think that's about as specific as I can be without running afoul of the forum rules, but do your research among the scientific literature, and here on the forum. I can quote some references to studies for you to read if you're looking for more information, otherwise, the search function is your friend.
