Comment 10 for bug 410846

You're not doing anything wrong :) The problem is as follows: the .ai file has some effects that were created by using special blend modes:
* Screen: for the highlights in the face.
* Multiply: for the dark parts of the jacket's lapel.
The author of this file did it this way for an unknown reason, I speculate that it was made that way because versions older than CS4 don't have support for gradients with transparency.

On the Inkscape side the problem is:
The PDF importer (which is the same as the AI importer) seems to correctly read the file, however Inkscape itself doesn't have support for objects with blending effects such as Screen or Multiply applied to them. This is a limitation of Inkscape which has problems rendering these effects on objects (correct me if I'm wrong) and you see those ugly blackish shapes. It can render them when applied to layers.

There are several workarounds:
1- Import the files to Inkscape and convert manually all the ugly black gradients to normal gradients (with many files it can get very boring, I know...)
2- Move the affected objects to a layer and set the Blend mode to Screen (lighten) or Multiply (darken).
3- Open the AI file and export it as PDF with flattened effects. I tested this method using the PDF-X3 profile and is the fastest if you're in a hurry. A couple of caveats: after flattening the file size increases a bit. Your attached file of 288 KB was blown up to 1.293 KB. I'm pretty confident that this size could be lowered down with some settings tweaking. The blended objects get converted to bitmaps which means that if you plan to upscale this too much pixelation will be visible.

If you have many files and don't have Illustrator I can convert the files with method 3 (supposing it has a batch conversion feature).