When can I neglect the voltage drop of diodes in a circuit?
It is depend on which diode model you are using to make calculations. There are three common diode models.
Exponential Mode:

Constant-Voltage Mode:

Ideal Mode:

So, as you see like all figures voltage drop on the diode is not just related to voltage. Current effects the diode too. If you want to answer when will you neglect the diode voltage drop, you should first select the which diode model you are going to use.
Figures are from: Microelectronic Circuits Adel S. Sedra &amp; Kenneth C. Smith (Table 3.1)