This も is the one of emphasis (definition 3 here). It has nothing to do with how many tests are yet to be taken. So in this case it means "as many as".
The definition given is not wrong, per se, but it's definitely lacking this emphasis. A better translation would be "I have as many as three tests this month", which, though correct, is not a typical way to express it in English. The assumed feeling is that maybe they usually only have one, possibly two tests a month; but this month is three.
I think to get the point across without using awkward, forced dialog, it would be better understood with visual or audible emphasis. Read as "I have three tests this month!!"; or spoken with clear emphasis on the word "three".