Word/phrase that means a series of problems of increasing severity caused by a small error

Snowball effect, which Josh beat me to, is the best choice.

Cascading failure can also be used.

A cascading failure is a failure in a system of interconnected parts in which the failure of a part can trigger the failure of successive parts. Such a failure may happen in many types of systems, including power transmission, computer networking, finance, human bodily systems, and bridges.

Also try the butterfly effect:

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

Or the domino effect/chain reaction:

A domino effect or chain reaction is the cumulative effect produced when one event sets off a chain of similar events. The term is best known as a mechanical effect, and is used as an analogy to a falling row of dominoes

Source: Wikipedia

Snowball effect may apply to you description:

• Metaphorically, a snowball effect is a process that starts from an initial state of small significance and builds upon itself, becoming larger (graver, more serious), and also perhaps potentially dangerous or disastrous (a vicious circle, a "spiral of decline"), though it might be beneficial instead (a virtuous circle). This is a very common cliché in cartoons and modern theatrics and it is also used in psychology.

(Wikipedia)

While I prefer snowball as an answer (provided by Tusher Raj and Josh61), another possibility is a "death spiral."

The situation or course of action of one who is on a path toward some sort of inevitable catastrohpic failure.

(wiktionary.org)

A Vicious circle is another option, not a single word sadly, but an often used phrase.