You don't need ; on breaks, in Javascript at least.
This code seems to run fine on my box; I just make a blank html and inserted the Script. This sounds like an issue with your CMS or how this jscript is integrated with your other code (if any).
That said, wouldn't it be simpler to write it this way? It seems a bit clunky to make an switch statement for 24 cases when you only have a few options;
Code:
<script language="JavaScript">
var today = new Date();
var hours = today.getHours();
if (hours >= 0 && hours < 6)
document.write("Midnight");
else if (hours >= 6 && hours <9)
document.write("Prime");
else if (hours >= 9 && hours <12)
document.write("Terce");
else if (hours >= 12 && hours < 15)
document.write("Sext");
else if (hours >= 15 && hours < 18)
document.write("None");
else if (hours >= 18 && hours < 21)
document.write("Vespers");
else
document.write("Compline");
</script>
Just a suggestion.. this will even work if you decide to expand to include the different prayers for each time.. just change the document.write string.