Automatically Numbering Labels

Yes.

And now the answer: Use merge to set up your labels. There is a button on the Merge toolbar that allows you to insert Word fields. The field {Merge Sequence #} which can be inserted using this button will put in a code which sequentially numbers each merged item. If you are doing labels you will have to add this code to each label in the sample sheet after the merge document is created, but when the merge is performed each label will be individually labeled in sequence.

What I have described does what you want. The important detail you may have missed is that BEFORE the merge you must insert the merge sequence # code on EACH label like this:
{Merge Sequence #}
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I tried this before I explained it initially and it works. Don't run the merge until you have inserted the Merge Sequence # on each label. Then every merged label (even if there are 1000s) will have the next number from 1 -> whatever on it.

You have clarified that you want the same information on each label, but the number to change from label to label.

I have successfully done this as follows:

Use Tools, Envelopes and Labels to set up your labels. When you have typed the information you want on each label, choose the "New Document" button to show the page of labels rather than sending them immediately to the printer.

With the page of labels displayed, use Insert, Field, Numbering to insert the code {LISTNUM} on each label where you want the series of numbers to appear. After you do this, you can choose File, Print Preview to see that each {LISTNUM} code has been changed to a number in the series. When you print the sheet of labels, these numbers will be printed on the labels along with the information you typed in the Envelopes and Labels dialog box.