I don't know much about FITS, but I thought it will be useful if I post this (and help avoid some confusion):

FITS is a rather complex format that can store different kinds of data in various ways.  Mathematica only supports certain kinds of FITS files.  It is likely that your files are not damaged, but Mathematica simply doesn't support this particular kind of FITS file.  I remember trying to read in some FITS files once, and after discovering this, I gave up on doing it directly with Mathematica.  You might convert it to an easier to read format with some other application, and then read the other format into Mathematica.

This is not a bug, and you're likely not doing anything wrong.  Mathematica simply doesn't support FITS fully, probably because of the complexities and the generality of the format.

(I did not actually download your files, I'm just sharing my previous experiences with FITS.)

OK, I know that.
I’m sending you two .fits files (A and B ) from LASCO:
https://www.dropbox.com/s/ltu1f0twu3gx5k1/LASCO_FITS.rar
I can open the file B.fts but I can't import the file A.fts. The two images captured by the same camera (C2), the A.fts is a decompressed image (Level 0.5) and the B.fts is a calibrated image (Level 1).