Question
The ADP2504 does not seems to provide the requested voltage (5V) WHEN A SLOW RISING Input is applied (bench power supply) at power up.Looks like the adp2504 is timing out and locking at the same voltage than the input in boost mode as the VIN IS 4V and VOUT SHOULD BE 5V.
Answer
How slow is slow?. It looks like its seeing as under-voltage lockout here. If so, try and use the EN function once you have good supply threshold.
Please ensure your Vin is < 4.3V see (PCN 11_0002 Rev. B) in the ADP2504 productline web page.
Alternatively:If EN is not software controlled. Try hardwire EN pin (function) i.e. connected to Vin with a zener with resistor to act as a voltage thresholdsensing between Vin and EN will do the trick.