According to the current FIDE Laws of Chess:
3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained
from moving to the square occupied by the king because they would then
leave or place their own king in check.
3.9.2 No piece can be moved that will either expose the king of the same colour to check or leave that king in check.
This means that one piece can move and give check and simultaneously uncover an attack from a second piece. There is no way for the moving piece to uncover two pieces attacking the king. Hence it is not possible for the king to be attacked by more than two pieces. However, it wasn't always so.
According to the 1985 version of the FIDE Laws of Chess:
9.1 The king is in check when the square it occupies is attacked by one or two of the opponent's pieces; in this case the latter is or are said to be "checking the king."
9.2 Check must be parried by the move immediately following.
The limitation to being attacked by only one or two pieces means that if your king is attacked by one or two pieces and you make a move which leaves your king attacked by 3 pieces then, in 1985, this did not count as being in check.
Once this was pointed out FIDE changed the rules to stop this.
This loophole was used in 1988 to construct this wild problem:
[Title "Robert Norman, CHESS Magazine June 1988, White To Move And Win"]
[FEN "1n5k/1r6/5K2/4Q1Pb/8/8/8/8 w - - 0 1"]
1. g6 Nd7+ 2. Kf7+ Nxe5++ 3. g7+! Kh7 4. g8=Q+ Kh6 5. Qg7#
Crazy as it sounds the move 3. g7+ by white is legal according to this loophole. The white king is attacked 3 times and so is not in check! The black king is attacked only once so is in check and so black must get out of check on the next move. This remains the case for all the rest of the moves.