How can I know the rated current of a 6v transformer?
It is because of rated current. Your transformer's core goes into magnetic saturation at some power, so you cannot exceed it. I guess you should change the load until the produced voltage on output is at rated voltage.

Are you sure it is rated at 6V AC (output)? All transformers when not in load produce little over rated voltage. On smoothing capacitor you would get (under no load): $$V_{dc (max)} = V_{ac} \times \sqrt{2}$$
If you get 8V, than I would expect it to be 5V rated transformer. (Under no load its voltage would be little above rated). Off course, losses on diode bridge should be calculated also. Usually they are ~0.6V per diode (on light load). You did not specified how do you rectify, half-wave (reduces 0.6) or full-wave (2x0.6V).

Because of high voltage drop you get under load, looks it is rated at small current, but it is hard to guess current without measuring.