It looks like you have a certificate in DER format instead of PEM. This is why it works correctly when you provide the -inform PEM command line argument (which tells openssl what input format to expect).
It's likely that your private key is using the same encoding. It looks as if the openssl rsa command also accepts a -inform argument, so try:
openssl rsa -text -in file.key -inform DER
A PEM encoded file is a plain-text encoding that looks something like:
-----BEGIN RSA PRIVATE KEY-----
MIGrAgEAAiEA0tlSKz5Iauj6ud3helAf5GguXeLUeFFTgHrpC3b2O20CAwEAAQIh
ALeEtAIzebCkC+bO+rwNFVORb0bA9xN2n5dyTw/Ba285AhEA9FFDtx4VAxMVB2GU
QfJ/2wIRANzuXKda/nRXIyRw1ArE2FcCECYhGKRXeYgFTl7ch7rTEckCEQDTMShw
8pL7M7DsTM7l3HXRAhAhIMYKQawc+Y7MNE4kQWYe
-----END RSA PRIVATE KEY-----
While DER is a binary encoding format.
Update
Sometimes keys are distributed in PKCS#8 format (which can be either PEM or DER encoded). Try this and see what you get:
openssl pkcs8 -in file.key -inform der