Yes. However, it is highly discouraged by immigration attorneys.
If you have been granted asylum, you should be aware that your grant of asylum may be terminated if, among other things, it is determined that you voluntarily availed yourself of the protection of your country of nationality, you have acquired a new nationality, or you are no longer eligible for asylum due to a fundamental change in circumstances. Similarly, your asylum status may be terminated if there was fraud in your application such that you were not eligible for asylum. Therefore, if your travel suggests that you may have re-availed yourself of the protection of your country, you should be prepared to explain how you were able to return to the country from which you fled.
If you have been admitted as a refugee, your status may be terminated if the U.S. Government determines that you were not, in fact, a refugee at the time you were admitted to the United States as a refugee. As a result, if your travel patterns indicate that you did not experience past persecution, that you did not have a well-founded fear of persecution on account of a protected ground, or that you persecuted others, you may need to explain your travel to that country to avoid losing your refugee status.
Speak to an experienced immigration attorney before filing Form I-131 for a Refugee Travel Permit to travel back to the country where you claim persecution.