It is not possible for one athelete to perform in two different weight categories in the same tournement, for the simple reason that weight categories don't overlap and there no longer is an "open" category.
A current list of weight categories is listed on wikipedia, and the same information is mentioned in a document on the IJF website. To quote the 2013 IJF rules, page 13:
8.5 Competitors may only be entered in the weight category to which they belong.
8.6 A competitor may not be entered in more than one weight category except for the
Senior Men and Women World Championships in which he can be entered also in open
Also, on page 24:
The competitor shall weigh-in wearing only underclothing, under the supervision of a
weigh-in official; the competitor may be requested to remove his underclothing to ensure
he reaches the minimum weight limit of the weight category in which he is entered.
On page 73, in discussing team competition, it is noted that judoka may not compete willy-nilly in any division above their own:
Each competitor is entitled to fight in his own weight category or in the next higher category.
The strongest evidence is on page 10, where the weight categories are described explicitly in terms of maximums and minimums. For example:
Middle weight - over 63 kg up to and including 70 kg