Adam was the eldest son of Sir Frank Forbes Adam, 1st Baronet and his wife Rose Frances Kemball.
Educated at Eton College and the Royal Military Academy Woolwich, Adam was commissioned on 27 July 1905 into the Royal Artillery. After a posting to India with the Royal Horse Artillery, he served in France, Belgium and Italy during World War I during which he attained the permanent rank of major (in November 1916) and was awarded the Distinguished Service Order, appointed an Officer of the Order of the British Empire and was mentioned in despatches three times.
After the war he attended the Staff College, Camberley from 1920 to 1921, and later held a number of successively more senior staff postings at the War Office. He also spent a period as an instructor at the Staff College between 1932 and 1935 when the other instructors included Viscount Gort, Bernard Montgomery, Philip Neame, Bernard Paget and Andrew Thorne. He was appointed Commander, Royal Artillery (CRA) for the 1st Division in 1936. He went on to receive in 1937 the prestigious posting of Commandant of the Staff College, Camberley, his first appointment as a major general. When Gort, a man of action but not particularly cerebral, was appointed Chief of the Imperial General Staff Adam was made Deputy Chief of the Imperial General Staff (DCIGS) "to be the thinking head whilst Gort provided the drive".
When Gort went to command the British Expeditionary Force (BEF) he wanted to take Adam as his Chief of Staff. The Secretary of State for War, Leslie Hore-Belisha refused the request on the grounds of maintaining continuity. However, in October 1939 he was appointed commander of III Corps which by February 1940 was crossing the Channel to join the BEF. III Corps was earmarked for Scandinavia in February, however by April the planned invasion was cancelled when Hitler invaded Denmark and Norway and the Corps remained in France. When in late May the BEF was ordered to evacuate, Adam was given the task of organising the Dunkirk perimeter (Major-General S. R. Wason RA took over command of III Corps). It is said that it was substantially due to Adam's leadership that the BEF was able to retreat behind a strong perimeter and leave France in relatively good order.
Following his return from France on 31 May 1940, Adam was appointed General Officer Commanding-in-Chief Northern Command, responsible for the defence of the coastline from The Wash to the Scottish border. It was during his year with Northern Command that he concluded that the army needed both more effective selection procedures and to ensure that soldiers understood the cause for which they were fighting. In June 1941 he was appointed Adjutant General, the second military member of the Army Council and a key role with responsibility for all personnel, administration and organisational matters. The role was of particular importance during the war years because of the need for the army to adapt its practices to meet the needs of a conscript army led by non-career officers. He set up a personnel selection department that drew up aptitude tests to establish recruits' psychological stability, combatant temperament, technical aptitudes and leadership potential. Under Adam's guidance, officer selection was no longer based on a simple interview by commanding officers, but carried out through a War Office Selection Board ('Wozbee') whose members, advised by psychiatrists and psychologists, oversaw various tests, especially those aimed at showing a man's initiative potential. Adam did not accept the traditional view that there was an 'officer-producing class', but believed that men and women of ability could be found in all parts of the community. Both these innovations met resistance, most of which was overcome.
Adam's championing of the Army Bureau of Current Affairs ('ABCA'), which produced fortnightly pamphlets on current developments to provide officers with material for compulsory discussion groups with their men. He and other senior officers recognised that the call of 'King and Country', which had been so powerful in 1914, was not enough for a more sceptical generation: a citizen army had to be encouraged into battle, not just ordered. However, the 'Theirs not to reason why, theirs but to do and die' attitude was still widespread almost a century after the Battle of Balaclava and the leftward swing in British public opinion during the war years that resulted in a landslide for the Labour Party in 1945 was blamed by some Conservatives on ABCA, a charge Adam considered 'absurd'. The ABCA discussion groups substituted the 'habit of rational argument for the anarchy of the barrack-room argument', he told the British Institute of Adult Education in 1945.
As the end of the war approached, Adam instituted a demobilisation system based on the 'first in, first out' principle and resisted attempts to repeat the practice in 1918–19 of giving priority to the needs of the economy, which had led to mutinies by long-serving men.
Adam was seen by Winston Churchill, amongst others, as being too radical and aroused the suspicions of more conservative generals. Churchill attempted to have him posted in early 1944 as Governor of Gibraltar but Alan Brooke, who had been appointed CIGS at the end of 1941 and who saw him as progressive, ensured Adam continued to hold the post of Adjutant-General while Brooke remained CIGS, which continued until the end of the war. Adam's influence on the conduct of the war was not only through his long tenure as Adjutant-General but also because he was one of Brooke's only two confidants, the two of them lunching regularly when both in London.
In retirement Adam became Chairman and Director General of the British Council, Chairman of the Executive Board of the United Nations Educational, Scientific and Cultural Organisation (UNESCO), Principal of the Working Men's College, and sat on the governing bodies of London University's Birkbeck College and Institute of Education. He remained a severe critic of the British educational system and in 1961 wrote that it would not be fundamentally changed until private education was ended.
He inherited the Baronetcy of Hankelow Court in the County of Chester upon the death of his father in 1926.
He married Anna Dorothy Pitman, daughter of Frederick I. Pitman in 1915; they had 4 daughters; Barbara (1917–2002), who married Sir Dennis Proctor, Margot (1918–1937), and twins Bridget (1927–2013) and Isobel (1927–).