No. Water dripping behind the gutters is not normal and there are three common causes to look for:
Spikes or screws that are driven through the drip edge of the roofing system when the gutters are installed.
Improper install of roofing system with ice and water underlayment placed underneath the drip edge.
Gutter system not tucked underneath drip edge.
If the issues above are not identified and fixed your home will sustain damage over time.