You're reply prompted me to try something different and it worked!
When I tried writing to a large range of cells using:

objXLWorkBook.Sheets("Sheet1").Range("A1:M3000").V alue = <my
multidimensional array of variants

It throws the 1004 error when the contents of a certain cell are too
big (somewhere between 200 and 1000 characters). But, if I write to
individual cells using:

objXLWorkBook.Sheets("Sheet1").Cells("A1").Value = <one element of my
multidimensional array of variants

I can enter in data as big as 6000 characters.

So, not sure "why" this works, but I'm using it! Thanks for the reply!