Incompatibility (noun)

1. The state of being incapable of existing or performing in harmonious or congenial combination.
2. The quality or state of being incompatible or inconsistent.

Origin:

Late 16th century from french incompatibilité or latin incompatibilitat- 'incapability of standing together', from in- 'not' + compatibilitat- 'capability of standing together'.

Examples:

1. The incompatibility of the two systems caused many problems.
2. The incompatibility of their schedules made it difficult to meet regularly.
3. The incompatibility of the chemicals resulted in a dangerous reaction.
4. The incompatibility of the two politicians' policies made cooperation impossible.
5. The incompatibility of their beliefs and values caused the relationship to end.