Yes, you can appear for both All India Law Entrance Test and Common Law Admission Test. But you need to meet all the criteria for both the exams in order to appear for it. Both the exams are conducted for admission to five year integrated course and LLM program.
http://law.careers360.com/articles/clat-vs-ailet
Check the eligibility criteria-
http://law.careers360.com/articles/clat-eligibility-criteria
And
http://law.careers360.com/articles/ailet-eligibility-criteria

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