I am by no means an expert in fencing, but from what I've been able to establish, this seems to be an over–simplification. The scoring system differs between individual and team competitions; and in an individual competition there are two stages, each with different rules.
First of all: an individual encounter is known as a bout, not a game. Second of all: it is possible to win a bout with fewer than five points.
If you're still interested, please read on!
The two stages in an individual competition are the pools and the elimination.
For the pool stage, the competitors are divided into groups and each player plays every other player in his or her group. Each bout lasts a maximum of three minutes, but the bout ends if either player scores five points before the three minutes are up. If neither player scores five points, the player with more points after three minutes is declared the winner.
The elimination stage takes a knockout format. In the first round, the player with the most points from the pool stage plays the player with the least points, and so on. In foil and epée fencing, each elimination bout consists of three periods of up to three minutes each. There is a one–minute break between each two periods. The first player to score fifteen points is the winner; if neither player scores fifteen points in nine minutes' play, the player with more points is declared the winner.
Sabre fencing is much faster than foil or epée, and bouts rarely last for three minutes. The first player to score fifteen points is the winner, and there is a one–minute break after either player has scored eight points.
In a team competition, each team has three players (plus one reserve). As in the pool stage of an individual competition, each player fences each of the three players on the opposing team, giving a total of nine bouts. Each bout lasts up to three minutes, as in individual competition, but in team competitions the points scored in each bout are carried forward into the next and the maximum total number of points increases by five each time.
For example, suppose the player from Team A wins the first bout by 5 points to 3. The maximum score in the second bout is ten points, so the second bout ends if either Team B's player scores seven points or Team A's player scores five (or after three minutes).
If, on the other hand, the first bout ends after three minutes with the score at 4–3 to Team A, the maximum points target in the second bout is still ten. The second bout will end if either Team B's player scores seven points or Team A's player scores six.
As there are nine bouts in a match, the maximum score in a team competition is 45. In sabre fencing one team will nearly always score 45 points, but in foil and epée it's not unusual to see scores in the low thirties.
To summarise: in the pool stage (of an individual competition), a bout can be won with fewer than five points, if neither player has scored five points after three minutes. In the elimination stage, you need to score fifteen points – but as in the pool stage, there is a time limit after which the player with more points wins. In a team competition, five points will win the first bout but subsequently it depends how many points have been scored in previous bouts.
My main source for this page was Wikipedia's Fencing practice and techniques page (Competition section). I also referred to the Fédération Internationale d'Escrime (FIE)'s Rules for Fencing , as provided on British Fencing's website (section t.30, point 3) – which has a good deal less detail but is a lot more succinct!
As far as I can tell, there is no difference between the rules or scoring systems in men's and women's fencing. The Rules for Fencing (referred to in the previous paragraph) do not contain the word "women".
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