I have heard from more than one person that all columns in the clustered index also show up in each non-clustered index on that same table, which is why you should keep your clustered index columns to the bare minimum you can get away with.
Is this true?
Yes and no*.
The clustered index (plus a uniqueifier if necessary) is the row pointer for non-clustered indexes (See http://msdn.microsoft.com/en-us/library/ms177484.aspx) So a smaller clustered index will mean smaller non-clustered indexes, all else being equal.
BUTif you have few non-clustered indexes, and you have a column that is almost always included in criteria for queries, you can gain performance if that's part of the clustered index at a small cost of space in the non-clustered indexes (indeed, you'd likely be moving that column from its own non-clustered index to the clustered index, so you could actually save space.)
Note: When I speak about changing the clustered index, I'm speaking of conceptually changing it preferably before any data is in the table, and definitely before moving it to production. Changing a clustered index in production can be difficult, to say the least. This is one of those areas where planning is key.
* This is ALWAYS the answer to any Yes or No question. Always.