Yes, you can. The classical A§E tests (phoneme detection, discrimination, identification) provide the option to use the test engine to create your own tests.
The detection engine presents a stimulus sound three times in a row and the listener should react if he/she hears it. The result is binary yes/no.
The discrimination engine provides two stimuli in a oddity paradigm. The background sound is replaced randomly by the odd sound and the listener should react to the odd phoneme.
The identification engine provides a closed set stimuli with a forced choice picture pointing task. A confusion matrix is built and binomial statistics are used to decide whether the listener has well identified all sounds presented.
You can use these engines to configure your own tests, with your own sound stimuli, your own pictures, the graphical layout and report style of your choice.