Yes, but you need to use a CSS Insert
Lets say you wanted to put the email.png (which is in the atahaulpa images folder) as a background to the 'Categories' widget. If you look at the generated code for a page, you see the widget looks like this:
HTML Code:
<div id="categories-3" class="widget widget_categories"><div class="widget-title"><h3>Categories</h3>
the ID in the first part has a number that increases as you use the widget so you can uniquely identify each occurance, but we will ignore it. To get the image you can use the CSS RULE
HTML Code:
{background: url("http://yourdomain.com/wordpress/wp-content/themes/atahualpa/images/email.png")
but now you must create the CSS SELECTOR.
If you just used 'div.widget-title' you would effect all the widgets. If you just use 'div.widget_categories' you would put the background behind the entire widget, so you must use a combination 'div.widget_categories div.widget-title' so your final CSS insert would be
HTML Code:
div.widget_categories div.widget-title {background: url("http://yourdomain.com/wordpress/wp-content/themes/atahualpa/images/email.png") }
hopw this helps