The British-English phrase (as) thick as two short planks means very stupid.

The humorous arbitrary comparison with two short planks gives emphasis to the adjective thick, meaning stupid.

—Cf. the synonymous phrase (as) thick as a brick.

These are, in chronological order, the earliest occurrences of the phrase (as) thick as two short planks that I have found:

1-: From April Noddy Is Not Dead, by the Liverpudlian journalist and author Frank Shaw, published in the Liverpool Echo (Liverpool, Lancashire, England) of Wednesday 31^st March 1954—Shaw listed Liverpudlian words and phrases relating to stupidity:

The shrewd citizen […] is quick to spot one pretending not to know. So, when a magistrate pompously assuming judicial ignorance, asked, “What is a jigger?” an irreverent Mary Ellen promptly replied: “Aw, cod on ye don’t know!”
She wasn’t as thick as two short planks.

2-: From Crack motorcyclist on speeding charge, published in The Winsford Chronicle (Winsford, Cheshire, England) of Saturday 29^th August 1959:

A 25-year-old international racing motor-cyclist, John Hartle, […] appeared at Nantwich Magistrates’ court on Monday on a charge of exceeding the speed limit in Chester-road, Barbridge, in a van. He pleaded “Not Guilty.” […]
P.C. J. E. Edwards alleged that Hartle travelled at between 40 and 45 m.p.h. in the restricted area at Barbridge, during which time he overtook two cars. When he asked Hartle why he had travelled at this speed, Hartle said “You never followed me. You don’t know your job, so be careful, be very b— careful.” When he warned Hartle in accordance with the Road Traffic Acts, Hartle said “You are as thick as two short planks.”

3-: From 4 Britons Program Love With Computer, by Mavis Cole, Chicago Tribune Press Service, published in the Chicago Tribune (Chicago, Illinois, USA) of Sunday 17^th December 1967:

London, Dec. 16—A computer is busy matchmaking here. It has tried more than 150,000 times in 15 months to make Britons say, “I love you.”
Four young Englishmen, three of them bachelors, started the dating project. […]
[…]
A student at Cambridge was so angered by one of his computer dates that he complained. A friend of his had been given the same girl, he said, and she was “as ugly as sin, as quiet as a mouse, and thick as two short planks.”

4-: From a letter by a certain Philip Baker, of the National Front ¹, published in the Evening Post (Reading, Berkshire, England) of Thursday 3^rd July 1969:

Most of the points raised by Basil Amps in his recent comments on the coloured invasion have been adequately answered often enough before. So it is rather difficult to understand why he raised them except as a peg for a personal attack not only on Enoch Powell ², but on the National Front and its members.
He uses such phrases as “those in the National Front who have half a brain,” implying of course that the majority are as thick as two short planks.

¹ The National Front is a far-right British political party, formed in 1967, with extreme reactionary views on immigration.
² John Enoch Powell (1912-1998) was a British Conservative and Ulster-Unionist politician, noted for his condemnation of multiracial immigration into Britain and his opposition to British entry into the Common Market.

5-: From an interview by Michael Hellicar of the Irish singer-songwriter Gilbert O’Sullivan (born 1946), published in the Daily Mirror (London, England) of Tuesday 25^th April 1972:

“People look at me on the television and they see me dressed up—you know, the flat cap, short trousers, or some other way-out stuff—and they think: ‘I bet he’s as thick as two short planks.’”

6-: From the portrait by Helen Dawson of Margaret McCorkell, a resident of Whitley Bay, near Newcastle upon Tyne, in north-eastern England, published in The Observer (London, England) of Sunday 14^th May 1972:

She describes herself as ‘easy-going.’ Is she ambitious for her children? ‘John says he’d rather they were happy dustmen . . . but I’d be secretly disappointed if they turned out as thick as two short planks.’

The following cartoon by Kevin Woodcock was published in the Evening Standard (London, England) of Monday 25^th March 1974—this cartoon depicts a psychoanalyst talking to a nurse about a man lying on the couch:

“You’d better give him a refund—after six visits, all I’ve got written on my pad is ‘nutty as a fruitcake’, ‘as thick as two short planks’ and ‘daft as a brush’!”