This is not related to the update (and the description you mention is incomplete, the full description is here http://wordpress.org/plugins/wp-linkedin/ :)
What happened is that you hit the throttle limit of the LinkedIn API, which means the [li_updates] shortcode is making too many requests. This is either due to you having a high traffic site without caching or some tests you are doing. That's why the [li_updates] is not working (and only this one)
Contrary to the other shortcodes there is no caching provided by the plugin for the updates (the reasoning being that updates change frequently and you want them up-to-date) but LinkedIn has a limit on the number of calls you can make per seconds and per day.
Possible solutions are either to cache the page or use your own LinkedIn application key and secret.
To do the latter you'll need to go here https://www.linkedin.com/secure/developer then create a new application. Once this is done modify your wp-config.php to include the following:
define('WP_LINKEDIN_APPKEY', 'your app key');
define('WP_LINKEDIN_APPSECRET', 'your app secret');
I might add some short term caching to that part too if this prove a common issue.
Lastly, the syntax you gave for the shortcode is wrong, in your exemple the
only_self argument won't be taken into account (it works because the default is
only_self="1") the correct syntax is
[li_updates only_self="1" count="7"] or
[li_updates only_self="0" count="7"]
Hope that helps