Is there any disadvantage to a static ip address?

[joe at actionline.com]

Is there any disadvantage to a static ip address?

My qwest dsl internet connection drops signal from time to time
(service is definitely not as advertised 99.99% uptime).

Also (qwest insists that is a separate issue), my web mail host loses
connection with an error message saying "ip address changed" several
times a day.

Someone has suggested that getting a static ip address should correct
some of these problems. qwest quotes $5.95/month for a static ip
rather than a dynamic ip address.

Is there any reason not to go for that?