I tried this function on Platinum today, but it didn't work just as always.  After the disappointment passed, I was determined to figure out why it doesn't work or why sometimes it works.  I am not sure if this is a issue for anyone still, but as usual the CW forum search did not find any threads on the subject, so please forgive me if this has been covered.
 
Typically, when you want to split a clip, you sent the Now Time where the desired 'slice' will occur and then select the offending clip/portion (unless already selected by defualt - Edit|Preferences|Customization|Editing|Clips| Selection after single split...), then go to Edit|Delete Special| and include Delete Hole as desired.  Except, this does not actually delete the hole left behind the deleted clip.  It either just deletes the offending clip or deletes both clips on either side of the slice (although the offending clip may be selected, the time range in the Rule is what is actually deleted) and the hole is removed.
 
First, before anything else, make sure your Click behavior under Options is not set to "Right Click Sets Now."   After the split, unselect all (I use CRTL+SHIFT+A) then Right Mouse Click on the offending section/clip.  You will now notice that the selected click is a little greyed out as opposed to a bright highlighted clip.  Also note the range of time in the Time Ruler.  Now when you select Delete Special and include Delete Hole, it will work as expected.  Another solution is to use the Select|By Time under Edit and plug in the start time and end time of the offending portion of a clip and then right click as above.  This might be a little more of a bother since you will have to find the exact starting and ending time (or "range") you wish to remove.
 
Just in case someone out there is still waiting for a solution to this dilemma.
 
Kind regards,
 
tecknot
 
edited to add missing step in bold.

I almost never have a single clip but sometimes have a section with multiple tracks that I need to delete (e.g. a doubled chorus or redundant interlude). Sometimes it will work but with a complex project often it won't. I'm not sure how your method would help me here since I need to select a bunch of clips by right-click dragging. I've also tried manually deleting the clips first, then the hole, but I can't get it to work reliably.

You can't delete a hole when there's nothing in it, insert an automation node, midi event, whatever and then you can delete the hole.
 

Have you tried it, Sander?  It should work with multiple clips as well as a single clip.  Just make sure you right click after selecting all unwanted clips.
 
Kind regards,
 
tecknot

gustabo
You can't delete a hole when there's nothing in it, insert an automation node, midi event, whatever and then you can delete the hole.
 

That is correct - it has been this way, certainly since X1, if not before.
 
I always insert a MIDI note for this purpose.

gustabo
You can't delete a hole when there's nothing in it, insert an automation node, midi event, whatever and then you can delete the hole.

What's stated here is my experience.  I delete sections all the time, but need a bit of something in them.

That's kinda weird, a hole is a hole, sometimes I just want to shift the markers back 8 bars. Nevertheless, the only reason I tried it is because it didn't work right with the clips there in the first place. I'll try right clicking the selection, tecknot. Doesn't that cancel/change the selection, though?

Sanderxpander
That's kinda weird, a hole is a hole, sometimes I just want to shift the markers back 8 bars. Nevertheless, the only reason I tried it is because it didn't work right with the clips there in the first place. I'll try right clicking the selection, tecknot. Doesn't that cancel/change the selection, though?

By default, right click brings up the pop-up menu and does not remove any selection (they remain intact).  All you need from that point is go to the Edit menu at the Menu Bar (atop).  Tried it myself in several projects and works like a charm.
 
As far as the delete hole as in an empty section, that can be done by selecting the range in the (Time) Ruler bar (or by selecting the start and end time as described in my initial post) and then go to the Edit menu and simply choose Delete or just press the Delete button on the PC keyboard.  You do not need to place anything in the empty section.  This is not to be confused with Delete Hole (though it seem like it should since your are in fact deleting a hole which really doesn't have anything in it, logically speaking).
 
Edit add. info.:  I have just tried to delete an empty selection or range in two selected tracks by using the right-click and Delete Hole method and it did in fact remove the empty/silence area in both tracks.  So, it does in fact work as expected without having to place data in the desired selection.    I did this by first taking note (not a musical value ) of the beginning and end time of the empty area.  Next, deselecting all selections (via CTRL+Shift+A) and then CTRL+click on the tracks containing the empty section(s) of silence - note that the empty range/selection will be the same across your selected tracks, you cannot select multiple ranges at once.  Now go to Edit|Select|By Time and enter the start and end time of the empty section.  Finally, go to Edit|Delete Special|Events in Tracks + Delete Hole, and, Voila! the section of "silence" is removed and the clips in the selected tracks are shifted to the left where the silence was selected.
 
Kind regards,
 
tecknot

I'm off to experiment, I'm not sure that I get exactly which part of your process is the missing magic step, but it's good to have this on hand when it's failing for me again, thanks!