Solution 1:

It's just a mechanism for expressing uncertainty or unimportance of the exact number. For instance, if someone says

We need to improve on-site safety. There have been five incidences in this month alone.

then you could be sure that there were exactly five of enough severity to count. However, if someone says

We need to improve on-site safety. There have been almost half a dozen incidences in this month alone.

then maybe it was five incidences, or four and a smaller incidence, or five and a small incidence. This indicates that the actual number is unimportant; it's simply big enough to be an issue, and helps to keep the focus on the real problem.

Also, by using almost or nearly the speaker suggests that this number might rise unless something happens to change the situation.

Solution 2:

Language is not simply about conveying raw information. It's about emotions as well. Compare:

There must have been four of five people fighting.

There must have been almost half a dozen people fighting.

In my perception, the second sentence is much more emotional and implies that the number of people fighting is impressive.