What we actually have here is a string of two suffixes. If we look at the example:
http://www.geyikmerkezi.com/fatih-urek-in-dun-hayata-go...-u..."Cenaze Töreni Neden Bir Kaç Saatliğe Ertelendi?"
[Why was the funeral delayed for a few hours]
we just see the '-e hali' or dative used on its own; this case seems to fit because there is an idea of extending from now up until that time.
With 'saatliğine', the third person singular possessive ending -i preceeds the -e suffix. This usage 'sounds right' to my ears but I find it hard to explain in rational terms. As a non-native speaker of Turkish, I get the sense that it makes it sound a bit more subjective and specific, i.e. the request is for a few hours to be devoted to the bags. I also get the feeling that it is slightly more polite. Undoubtedly, native speakers will have deeper insights than me into this usage.