I think the claim that NATO served primarily to keep "the Germans" down is false, and it is not a widely accepted justification; more precisely, NATO served to bind West Germany into a coherent Western alliance, thus preventing West Germany to turn into a "third pole" in an increasingly bipolar world. That means less than "keeping the Germans down" but "keeping the Germans close" (even if initially the motivation to keep a German remilitarization "down" was clearly part of the idea).

More precisely, this statement ("...and the Germans down") has to be seen not as a manifest policy realized through official regulations but as an interpretation of intent.
While there are authors who recently tried to make the point that a good deal of US cold war policy in Europe was focussed on how to ensure that nuclear non-proliferation would remain true for Germany in particular (cf. Marc Trachtenberg, A Constructed Peace), and that Soviet policy especially from 1955-1965 was fueled by fear the West Germans would gain their own nuclear weapons, this has not gained much traction, mostly because the post war period is massively more complicated the closer you look, and there are few single overarching factors or policies (eg like "keeping the Germans down") that dominate political decision-making through the entire period.
The post war era in Europe is characterized by various changing phases of economic and political dynamism which could not be easily predicted even from one decade to the next: in 1945 no one could foresee the simultaneous quick re-rise of the (West) German economy, the shifting local foci of the various cold war crises (Berlin, Korea, Poland, Berlin again, Hungary, Cuba, Berlin again, Cuba, Laos, etc etc), the fast decline of the British Empire (cf decolonization, Suez crisis) and the shaping of NATO, all in the period 1950-1962; neither did the United States realize the fast rapprochement of Germany and France, which in the early 1960s lead to the sudden realization that West Germany had developed from an occupied vassal state to an internationally relevant economic power whose opinions and policies suddenly mattered (West Germany surpassing the British economy in GDP in 1960).
The same was again true for the next decade: The fast development of a French dislike for US hegemony in Europe (with France even leaving NATO), Franco-German cooperation and at the end of the 1960s the switching in (West) Germany to a left-moderate government opposed to continuing the arms race and preferring a Western switch to a detente policy kept changing the ground on which international policy makers had to calculate their decisions; while the United States got bogged down in Southeast Asia, and proxy wars raged through Africa, Europe was trying to calm international waters through active diplomacy, which had feedback effects on American policy on Europe and NATO.
And in the late 1970s, few felt that the next decade would see yet another swing of the pendulum, where the cold war got colder again (Afghanistan etc), and the stationing of new nuclear weapons systems in Europe would become a tricky issue with America's own allies; similarly, both Europe and the United States were actually unprepared for the sudden collapse of the Soviet Union, East Germany, and the various communist satellite regimes by the end of that decade.

By the early 1960s, the American view of West Germany had, while still fed with suspicion from the previous world wars, fundamentally changed: on one hand, West Germany should not be trusted to determine Western military security policies, or attain its own nuclear weapons; but on the other, West Germany's economic input and technological development became more and more relevant, accompanied by appropriate say in international diplomacy. A good example here is the attention the United States gave to both (always West) Germany's legal guarantee that it would not develop ABC (atomic, biological and chemical) weapons, but there was a loophole that did not preclude West Germany from getting such weapons from elsewhere. France had begun developing its own nuclear capabilities, and even signed a nuclear technologies cooperation agreement with the Germans (which came to little, but was an important diplomatic card to play against the US, forcing Washington to keep an eye on developments in Europe and consulting Paris and Bonn to ensure they would not stray too far -- which they didn't.) By 1960, German chemical corporation Nukem, for instance, had developed an ultra gas centrifuge which could drastically reduce the cost of enriching weapons grade fission material. Such technological capabilities required the United States to ensure that the Germans would remain "happy" in the alliance, and bound US protection to Central and Western Europe: the dependent ally had suddenly a way to pull on the leash it was kept on. While West Germany was keen to prove that it was a reliable Western ally, and had no intention of actually developing WMD capabilities without Washington's approval, the period 1957-62 saw a (short) phase where both Washington and Bonn were considering giving Germany a "finger on the trigger" (cf. MLF discussion). In other words, at that point West Germany was more or less openly displaying that, if left unprotected by the alliance, it would be both capable and willing to develop its own WMDs.

NATO, therefore, at this point does not keep "the Germans down" but it does keep "the Germans close". And that worked both ways.

As for conventional military capabilities, West Germany was fast developing its own again under the umbrella (and with the approval and knowledge of) other Western governments and NATO. The insurance that West Germany would not prepare for a war against its Western European allies was indirectly achieved through the European Coal and Steel Community (forerunner to the EEC, EC and EU), which guaranteed that any massive military buildup of the Germans could not be kept a secret (through a jointly controlled coal and steel market).

I hope this makes sense, in a nutshell.