Yes, it could be ... but ...

1) Jasper to Banff is only 180 miles (288 kms) so it could be done quite easily in one day if a person wanted to do it that way ... especially if you aren't carrying much gear. That would be the least expensive option.

2) If you did want to break it up, there are places to stay along the way. There's a hotel at the Columbia Icefields, a collection of cabins at Saskatchewan River Crossing, a variety of accommodation choices at Lake Louise, and some between Lake Louise and Banff. And there are the hostels.

3) If you went with the hotels and cabins, they are EXPENSIVE!!! $200+ per night is not uncommon. See Icefield Chalet info: http://www.brewster.ca/Columbia_Icefield/Chalet.asp

4) If you opt to go with the less expensive option, hostels, book ahead ... do not count on being able to pull up to a hostel and get a room!! If you want to make use of the hostels between the end of May and the end of August, you will likely have to book a good 6 months to a year in advance (depending on the hostel). Outside of those dates, most of the hostels require at least a 5 day in advance reservation, and some may close for the season. You can check this site for further information:
http://www.hihostels.ca/en/index.aspx?sortcode=2.3