Is deleting secondary files a bad idea?
There are many considerations when determining the right number of files in a filegroup--Determining the proper number and layout of files in this filegroup is a separate question, but one that should not be ignored. File layouts for a database can be non-trivial, and the topic of adding/removing files from a filegroup raises lots of tangential discussion points.

That being said, I'm going to assume you've already looked at those things and come to the conclusion that you must remove some files.

If you choose to remove data files from a filegroup, the first step is to run DBCC SHRINKFILE(logicalname,EMPTYFILE). (If you don't do this manually, the GUI will do it for you--I recommend against doing this sort of thing via the GUI.) This will evacuate data pages from this file and move them into another file in the filegroup that has room. This is done page-by-page, and is an online operation. However, just like any file-shrinking, it's going to cause all sorts of nasty fragmentation. 

You probably want to run index maintenance after you remove the files. 

Before you begin, you will  also want to consider which file(s) you want to remain at the end. I would go ahead and pre-grow those files so that you're not trying to autogrow FileA while trying to shrink FileB. Or possibly, I would just create some brand-new files and get rid of the existing 16 files.