STAR*D showed that higher dosages of antidepressants (of course up to the FDA max) for most patients usually does increase efficacy of the antidepressant. If a patient has severe depression and I start then on an antidepressant, while in inpatient, I tended to increase it faster than the 1 week-norm for pretty much all of them. I never raised it faster than every 4 days. As mentioned above you risk serotinergic syndrome. Warn the patient of what you're doing and offer then the choice to do the norm or go faster with higher risks but hopefully (but possibly not) faster improvement (but higher risk of side effects).
What I've seen and don't understand at all is after starting and rapidly titrating an antidepressant, switching to a new agent if the patient is just as depressed after 5-7 days.
Likewise I don't get the psychiatrists (or really any other doctor treating any disorder) that don't well-catalog what's already been tried and the dosages. E.g. pt tried on Wellbutrin, they were stopped but the doctor didn't write down in the notes why, the effects good or bad, and the duration the patient was on it.
So often times I get a new patient and even if I have the prior doctor's notes I tell the patient I can't make heads or tails if they really had an adequate trial.
On my own notes I keep a section in the patient's history of "Psychotropic Medication Trials" that I don't write into the history section until the pt's had an adequate trial (e.g. Antidepressant-1 month of med at the highest FDA dosage of highest tolerable dosage for the patient.
Here's a cut and paste from one of my patients.
Belsomra: Helped with sleep at 15 mg. Denied side effects.
Buspirone: "made me sick." even at low dosages first few days.
Carbamazepine: "sick to my stomach, bad no appetite, numbness, muscle aches, nausea, not being able to sleep."
Desvenlafaxine: "Worked great," denied side effects at maximum dosage.
Duloxetine at 60 mg by mouth: some improvement. Doctor that prescribed it moved out of the area so she wasn't able to continue it.
L-Methylfolate: Improved depression. Worked better at 30 mg daily vs 15 mg daily. No side effects.
Methylphenidate: Made XXX more anxious
Melatonin 30 mg in one night: No benefit.
Olanzapine: 20 mg-IBS sx are gone, appetite is up but so far this is what she wants.
Paroxetine: Worked but only until she became an adult then it stopped working.
Propranolol: Didn't help at lower dosages but at higher dosages it lowered her BP too much.
SAM-E: no benefit with mood or joints despite being on 1500 mg daily for 1.5 months.
Viibryd: Felt no difference on the starter pack 2 week trial
Wellbutrin: "Made me go completely panic attack." even at small dosage first few days.