I connected a commercially available LED to a circuit in series without any resistor with a telephone line in India. The voltage available from the telephone jack is 48-54V DC with Positive Grounded and the max current is limited to 40-60 mA by the controlling cards. Why am I able to use the LED without fail? My question is how did the LED withstand 60mA current?
An LED is not a quickblow fuse; it will not fail instantly if overdriven.
Failure isn't caused by current directly, it's caused by high junction temperature either destroying the junction or melting the bond wires. So if your current is limited to 60ma, that will heat up the LED slowly, probably at a rate where its leads will conduct away much of the heat. You will have shortened its life, and possibly distorted the colour, but it will most likely run for hours or days at this level of (limited) overcurrent.
60 mA is not a lot of current for certain LED lamps. The high powered ones are designed for 700 mA or more. Of course they have to be designed to handle the heat load. The power supply being current limited to 60 mA means that it behaves like a current source (voltage changes to meet the conditions necessary for 60 mA). An LED tied directly to that same power supply but with the current limiting set to a higher value would have killed the LED's.
An Ideal voltage source has zero output resistance (i.e it is very "stiff") and an ideal current source has Infinite output resistance (i.e. it is very "soft") and has high voltage compliance.
I connected a ... LED ... without any resistor with a telephone line
The resistors in the circuit are at the telco end.
The current drawn may be less than 60 mA depending on the standards that the telco's equipment complies with.
The maximum total current which may be drawn from the line while in the idle line signalling state is 2,5 mA
See European Technical Standard ETS 300 001 "Attachments to Public Switched Telephone Network (PSTN); General technical requirements for equipment connected to an analogue subscriber interface in the PSTN" Section 2.2.1 (GB) 4, Page 16.