Yes you can. It boggles my mind that anyone would hire someone else to follow the step by step tutorial created by Staff and linked to on the bottom of their Admin page here http://learn.wordpress.com to set up a free hosted WordPress.com blog but they do. More to the point, most of them and those they hire never read this entry and comprehend fully what's in it. http://support.wordpress.com/com-vs-org/ Please do that now.
More galling still is when someone is hired to set up a free hosted blog which we Volunteers can do in not time flat and then spends days here posting threads and pumping us Volunteers for answers to questions that are already in the tutorial and support documentation and gets paid for doing that. But it happens.
You cannot build or use your own custom theme built from scratch at WordPress.com.
You cannot upload plugins. http://en.support.wordpress.com/plugins/
The only themes we can use are those found here > Appearance > Themes
This is a multiuser blogging platform. All blogs wearing the same theme are using the same underlying template and we cannot access the underlying files and edit templates or themes. http://en.support.wordpress.com/themes/editing-themes/
If you have CSS editing experience the upgrade will allow you to stylize the appearance of themes you find here > Appearance > Themes, but will not allow you to change the functionality, by editing the underlying template. It does not allow you to remove footer links to the theme designer or to WordPress.com. Also note that CSS is theme specific, so you cannot use the CSS stylesheets from other themes with the themes at WordPress.COM. http://en.support.wordpress.com/custom-design/custom-css/