Could it be your transformer was designed for pulse operation and the current rating given is meant a peak rating rather than continuous?
Or perhaps you mistook power for current?
Are you also sure you aren't mistaking the transformer's current rating for it's output current rating? A transformer's current rating is the maximum current the transformer's coil can handle and NOT the output current.
So a 1:10 transformer rated for 1A can handle 1A on it's primary side or secondary side. Since it's a step-up transformer, this means that the side that will draw more current in operation is the primary, therefore the secondary side can only output a maximum of 0.1A.
If it was a 10:1 transformer rated for 1A, then the secondary would always have more current in it than the primary. THerefore this transformer's secondary can output 1A and the primary will only ever draw 0.1A.