Well, you need to % with 7 (number of days in a week)
var weekday=new Array(7);
weekday[0]="Nedelja";
weekday[1]="Ponedeljek";
weekday[2]="Torek";
weekday[3]="Sreda";
weekday[4]="Četrtek";
weekday[5]="Petek";
weekday[6]="Sobota";
var dan = weekday[today.getDay()%7];
var dan_1 = weekday[(today.getDay()+1)%7];
document.getElementById("day1").innerHTML = dan_1;
var dan_2 = weekday[(today.getDay()+2)%7];
document.getElementById("day2").innerHTML = dan_2;
var dan_3 = weekday[(today.getDay()+3)%7];
document.getElementById("day3").innerHTML = dan_3;
var dan_4 = weekday[(today.getDay()+4)%7];
document.getElementById("day4").innerHTML = dan_4;
var dan_5 = weekday[(today.getDay()+5)%7];
document.getElementById("day5").innerHTML = dan_5;
For example: if today.getDay()
returns 5, next 5 days is 10 which does not exist in your array. When you % 7, you get back 3 (next week)