Hi there,
It seems that your baby quail are not getting the nutrition that they need. Usually the chicks use up all the nutritional reserves from the yolk after 3-4 days. So on the 4th day they start dropping due to starvation. I did an experiment where I had two batches of quail, one with a heat lamp and one without. Many from the batch without the heat lamp dropped dead after 4 days. What the white light from the heat lamp helps do is keep the chicks awake longer so they are stimulated to eat more. The chicks can move freely under the light or away from the light so they are able to self-regulate their temperature (Damerow 3).
I suggest getting a heat lamp and a thermometer (ASAP) and place the heat lamp far enough where the thermometer reads 95-100 degrees Fahrenheit. Measure the temperature at the height of the chicks, at least 18 inches. As the chicks get older, gradually decrease 10degrees per week (Ernst 5). I can't recommend exactly how many inches away from the height of the chicks because that would depend on the wattage of the heat lamp.
Lastly, light affects chick growth rate. Even if you dim the light, the light should still be bright enough to see whats going on in the brooder (Damerow 3). Below, I added the references that I used.
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Jazmine
Damerow, Gail. "Raising Chicks: Chick Brooder Temperature and Light Requirements." Mother Earth News. Ogden Publications Inc., 1 May 2013. Web.
Ernst, Ralph A. "Raising and Propagating Japanese Quail." UC Davis Depatment of Animal Science. University of California, Davis, 1 Oct. 1978. Web.