The main problem with submarines is that it is not possible to communicate with them while they are submerged. In particular, the submarine can receive but not transmit ELF signals and nothing else really penetrates sea water, so if you want to immediately know the status of a submarine you would have to transmit an order to surface and report in (doing so is rare because it is very easy to detect the submarine when it surfaces). If a country relies exclusively on submarines it will be vulnerable to a surprise "disarming attack" -- the submarines could all be destroyed while at sea, and then the country could be attacked with a first strike and would have no chance to retaliate. Exclusively relying on the submarine force will also allow the other side to devote more of its resources to anti-submarine warfare.

One reason to maintain an ICBM force is that the ICBM silos protect the missiles against anything less than a direct nuclear strike; by placing the silos far from urban areas you force the opponent to expend many nuclear weapons attacking individual silos, which greatly increases the cost of a first strike. If you were wondering where bomber forces fit in, maintaining both ICBM and bomber forces creates a "synchronization problem" -- both must be destroyed at the same time to prevent retaliation, and by strategically placing the air fields and missile silos a country can make a synchronized attack more difficult (the attack will be detected long before anything is destroyed, but one of the two will have more time to initiate the retaliatory strike). Bombers also force the opponent to expend significant resources on air defense, resources that could otherwise be devoted to offensive weapons. Another reason to have a bomber force is that bombers can be recalled after being ordered to strike; missiles cannot be recalled once launched and even if they could be the other side will detect the launches and immediately retaliate.

The US and USSR developed an entire theory of the nuclear triad during the cold war, and both reached the same basic conclusions about the need for all three legs.