Is there an allocated time period in Ray Bradbury's novel? I'm not too sure if there is a timeline that led up to the events in the book, so the time in which it is set is unclear to me.
Within the book itself there is no definitive date but with a little creative analysis we can we can date the book as being set in the year 2053 (or maybe a couple of years afterwards).
Within the text of Farenheit 451, Clarisse states that her uncle was arrested for an unusual crime;
That particular story was told in another Bradbury short story called 'The Pedestrian' which contains a reference to being set in the year "A.D. 2053" (coincidentally 101 years after the date of publication).
In an interview with 'American Author' on the occasion of his 80th Birthday, Bradbury clearly states that 'The Pedestrian' is a direct prequel to Farenheit 451.
Additionally, the novella from which 'Farenheit 451' was adapted (namely 'the Fireman' originally published in Galaxy Magazine in 1951) explictly states that the year is
Given Bradbury's apparent predilection for placing his futuristic books a hundred and one years in the future, the figure of 2053 makes perfect sense.
The novel originally started out as short story called Bright Phoenix that Bradbury wrote in 1947. He later expanded it to a novella titled The Fireman (published in Galaxy Sci Fi 1951) and finally into the novel Farenheit 451. See article. Interestingly, Bright Phoenix wasn't published until the May 1963 edition of the Magazine of Fantasy and Science Fiction.
The quotation from Farenheit 451 referenced in Mistah Mix's answer is (according to Note 1 in this Wiki article) ambiguous and open to interpretation. That same note however also states that the novella sets the time in October 2052.
The full text of the novella is available here. The formatting is 'unkind' but the beginning of the novella is explicit regarding the date -