Yes, constructors are allowed to throw exceptions.
However, be very wise in choosing what exceptions they should be - checked exceptions or unchecked. Unchecked exceptions are basically subclasses of RuntimeException.
In almost all cases (I could not come up with an exception to this case), you'll need to throw a checked exception. The reason being that unchecked exceptions (like NullPointerException) are normally due to programming errors (like not validating inputs sufficiently).
The advantage that a checked exception offers is that the programmer is forced to catch the exception in his instantiation code, and thereby realizes that there can be a failure to create the object instance. Of course, only a code review will catch the poor programming practice of swallowing an exception.