Instead of "splitting" your Title, use the Duplicate command from the Titler, to create a duplicate of that Title, and name is differently, say "Title_Name" 02. Place both Titles on the same Video Track, adjust the Duration, as needed with two Titles, and apply one Animation Effect to one, then another Animation Effect to the other.
Another workflow would be to skip the Animation Presets, and build those via Effect Keyframes, on say Opacity. That is not quite so simple as the animation Presets, but allows much more control.
Thank you so much for the quick reply, this seems to work great, however I think I'll take your advice and switch to keyframing. I really appreciate it!
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One thing, that is not all that readily apparent is exactly how Titles work. One can have any number of Instances of a particular Title. That, however, is not the same as having multiple Titles (they can be the same, or modified), and those Instances are linked - any change to one, affects all the rest. By Cutting a Title, one has just created two Instances of the same Title. The Duplicate (from Titler), creates two physical Title, though in your specific case, they appear to be the same.
For more info on Titles, this article might be useful: http://forums.adobe.com/thread/1131177?tstart=0