Dostoyevsky's neighbor was Alexander Barannikov, a revolutionary and terrorist, who participated in the murder of the Russian chief of police and Moscow bombings.
The story was first discovered by Viktor Shklovsky, our great literary scholar, who included official letter about the search in his novellette published in 1933.
Later this story was further investigated by Igor Volgin, Russian historian.
Joseph Frank's biography tells the story:
other events were taking place in the very building where the
Dostoevskys occupied apartment 10. Apartment 11 was actually a small
rooming house where individuals could rent single accommodations.
Sometime before midnight on the twenty-fifth, the police entered that
apartment and carried out a thorough search of one of the rooms in the
presence of witnesses. Its inhabitant had been arrested elsewhere
earlier that day, and, although he carried a false passport, there was
a wellfounded suspicion that he was a member of the executive
committee of the terrorist Narodnaya Volya
After the search police made an ambush and the next day, January 26th, another revolutionary was caught in this apartment, Nikolay Kolodkevich.
Dostoyevsky's wife Anna doesn't mention this in her Reminiscences, but both Shklovsky and Volgin try to cast doubt on her account of the events. Anna mentions her husband "moving a heavy bookcase” which led to him rupturing his artery. But in her original draft it is him "having lifted a heavy chair". Was she making things up?
Volgin cites police report of the search and gives the document's code number in the State Archive, so I guess it should settle the question of the possibility of the event.
After Tsar's assasination Barannikov was actively preparing himself and his family for the death penalty. However, his actual sentence was 20 years of katorga. That unexpected turn crushed him. He died in a year.
Shklovsky's novellette was published in the Soviet magazine Krasnaya Nov, 1933, December, p. 151
Volgin's book - [Volgin I. Posledniy god Dostoevskogo : istoricheskie zapiski [The Last Year of Dostoevsky: Historical Notes]. Moscow, AST Publ., Zebra Publ.] Chapter XX
Joseph Frank - "Dostoevsky: The Mantle of the Prophet, 1871-1881" - 2002 by Princeton University Press, pp. 744, 745