はい、できません。両方送ると、最新の結果だけが使われます。
審査官用のマニュアルに次のようにあります。
In terms of designated language test results, officers will NOT select the highest score for each language ability from a variety of test score submissions. It is always the most recent group of tests that will be considered by officer as it is the most current assessment of the applicant’s four language abilities.
http://www.cic.gc.ca/manuals-guides/english/op/op06e.pdf
CELPIPもIELTSも、どちらも移民申請で有効です。