Both the USA and USSR did have limited Anti-Ballistic Missile systems that were in place. These typically involved using one nuclear weapon to shoot down another — the "defending" missile would go up and detonate a relatively low-yield weapon fairly high up, hopefully in the missile path, with the idea of damaging the missile. The ABM Treaty of 1972 limited the number of such systems each country could have.

The US system, Safeguard, cost $5 billion dollars and was mothballed 5 months after going online. There were a lot of issues with the system.

The USSR fielded such systems in the 1960s. Their technical reliability seems somewhat dubious.

Any ABM system of this sort suffers from the problem that it is much harder to defend than it is to just increase the offensive attack. So if you have a system with 100 interceptors, the opponent just has to either target more than 100 missiles (and both sides had thousands), start MIRVing (adding more warheads per missile), or start adding dummy warheads. Much less more direct means like radar "blinding" (have one of your nukes go off just before reaching the limits of the interceptors), targeting the interceptor support systems (like radar), and so on.