The few pension files I have looked at include two cases in which someone who received a pension was not eligible for it.
Ambrose Baker was killed on 30 September 1864 at Peeble's Farm, Virginia. His wife, Mary Ann Barger, received a pension until November 1878, although she had "continuously cohabited" with John Deal since about 1868, and many (but not all) people regarded them as married.
Frederick H Lehman died on 10 January 1864. He had married Lizzie Croskey [?] on 11 Jan 1858, by Reverend Jos. Kennard. She received a pension as his widow from January 1864 until March 1890. Unfortunately, under the name Cora E Lehman she was married to Michael Gillen on 23 Sep 1867. And after he died, she married William T McGuire on 26 Feb 1884. Since she had repeatedly sworn that she had not remarried, a pension examiner found that she had knowingly committed perjury four times each year and had illegally drawn over $2300 from the US. He recommended that criminal charges be brought against her. The abbreviated file I received ends there.
Garret Hurst's widow was investigated, but because of a fraudulent execution of documents by her attorney, George W Ford. Payment of her pension was suspended during the investigation, but the special agents recommended that the suspension be removed.
Also, John A Langerbartel's widow was investigated because of some question about her original statement. The agent recommended that payment be resumed.