Yes, in fact this is quite common. It's expected that board members will play a key role in raising funds for the organization, especially at the beginning when you can't afford to hire paid development staff. That will often mean that board members ask others to contribute money. A board member who is able to say, "I and other members of the board have already given a total of $x to this cause" is in a much better position to convince others to act similarly.
Of course, you don't want to make your board so exclusive that only the rich can join. While many organizations set a suggested board donation amount, many also create a sliding scale as appropriate to bring in a broad range of community members.