I've been looking into this for a friend and going through forums to investigate this error.  After you've checked your database connection details, I find what FisherC said below is the most probable cause for the error.

FisherC says:

I have the same problem, but I'm not sure for the same reasons.  I installed the standard 1.5 installation package. I have uninstalled and reinstalled once just to make sure everything was OK. Basically after I use my website (using the default Joomla welcome template with NO modifications whatsoever) I start to get the "cannot connect to MYSQL." I contacted my webhost, and the problem is the number of MYSQL database queries. It goes up to about 200,000 in the first few minutes of use and then after that every page load is a few hundred to few thousand queries (per page). The webserver limits me to 50,000 queries an hour and I am blowing past that quickly with just 1 person (me!) on the site so they just refuse the connnection after a while.  Obviously something is not right. I am searching the forums for an answer and if I don't find one I'll repost. I just thought I'd share this with you in case this your problem too.

Source: Joomla Forum: http://forum.joomla.org/viewtopic.php?f=433&t=198257

My Solution

I have over 20 joomla websites on my virtual server and I haven't had this problem.  

1. I setup a website with about 20 3rd-party components installed, 15 modules and about 15 plugins (excluding the core modules/plugins) and managed to recreate this error.  
2. I disabled half of these but still got the error.  
3. Cleared the expired cache but still got the error.
4. I then uninstalled half of these (including Kunena and Agora forums), specifically database intensive plugins and the error happened less...
5. I then went through the menus and installation screen to get rid of anything that was not visibly used on the website.  The website was working fine and I was able resume my click-happy habits.

It's a given that in my test, there was only 1 user (me).  There are reports though that some people set up websites with NO 3rd-party items whatsoever... For those of you with this issue, I'd double-check your website host allows Joomla and if they don't, look for either an upgrade or a different host.

If your website has been LIVE for a while and it's returning this error despite having little installed, then check your logs to see if anyone has been trying to hack your website.  There are a number of components I have found that even with captcha features, still get hacked with viagra emails and what have you (eg. JoomlaBook).  If you don't want to go through the logs, try setting up a test joomla site with a different database and user and see if that behaves the same way.

Virtual Hosts

I used to have an account with 1&1 (oneandone.co.uk) and I would strongly urge Joomla developers to avoid them like the plague.  They have been less than helpful and when I only had one Joomla site with them, they still would block me thinking I was trying to hack into their servers because of the number of queries Joomla does and the number of times I refresh the page and upload files.  Why they think that developers should not be able to send a 1000 queries a minute or why a developer has to refresh the page, make a change, upload, etc.  well it's beyond a joke and no one should waste their time with such companies.

 Keyboard and mouse shortcuts are combinations of keyboard strokes and mouse clicks that you can use to perform specific actions. Knowing some of the Google Chrome shortcuts listed below can help save you time.

 This is in the case that you have only one optical drive which is a minimum of a CD Writer and you want to make a copy of a CD.

1. Insert original CD in the drive
2. In the CD Drive dialog box, click CANCEL
3. On the desktop, double-click on MY COMPUTER
4. Right-click on the CD drive and select OPEN
5. Go to EDIT > SELECT ALL > EDIT > COPY
6. PASTE to a temporary folder
7. Select all the files, right-click on them and select SEND TO > DIRECTCD

The jury's still out on this one.  I have noticed a slight improvement in my page loads but will have to carry out proper tests on the weekend.

Instructions to a hard-coder

The Google Chrome Frame (or ChromiumBrowser) is quick to enable.  Simply ask your webmaster to put the following meta tag in between the head tags of the page:

Old Server

mysqldump -u root -p wikidb > wikidb.sql
tar -cvf wiki.tar wiki   ;this is the wiki folder on document root

New Server:

create database wikidb;  this is inside mysql, Note that both mysql versions should be same.
grant create, select, insert, update, delete, lock tables on wikidb.* to wiki@localhost identified by 'YourPassword' ;

MediaWiki Upgrade

copy all the new files to wiki folder and then run php update.php from maintenance folder after updating AdminSettings.php

More Information

http://www.mediawiki.org/wiki/Manual:Moving_a_wiki

Source: http://forum.joomla.org/viewtopic.php?f=431&t=401086&p=1693742#p1693742

It's been difficult to find an easy solution to hide menu items from registered users. Finally, I found a solution that works 100% for me using the MetaMod tool without using any template hacks. The instructions on the metamod page are very vague so I wanted to make it bit more in-depth for this specific use. I'd love to hear if this helps you. Let me know if you find any mistakes and I'll be sure to fix asap.

Create a working Joomla Template from your Photoshop PSD design in seconds.

 Shame the soundtrack is so corny but otherwise I liked this video...

 Well we're all happy that the iPlayer from the BBC has come to PS3 in Europe but I'm less impressed about the multi-audio output.  My setup is that the sound goes from the Playstation to the Sound System via Optical Out.  

I found that if you do enable the multi-audio output then you can play music again.  

What a bummer you can't play music and use anything else anymore.  I can still play some games and select my own music... but they're dumbass PSN games that iPhone users would have got for a pound or what have you...

This is what I had to do to be able to add another template to the dropdown in the Kunena configuration panel:

1. Download your site from your server to a copy on your local disk (hard disk, flash drive, etc)
2. Make sure that the \components\com_kunena\template folder is on your local disk
3. Make a copy of the default folder inside
4. Rename this to the name of your new template (avoid any spaces)
5. Download the new template for the Kunena forum

1. Download the Template package.
2. Open your FTP Connection and go to \components\com_kunena\template\
3. Extract the folder "Template Name" into this directory ^
4. If in the package the folder "lib" contains, extract the files in the directory \components\com_kunena\lib\

 I've been told of two ways of running a downloaded .RUN file.

Note: Bear in mind that the following is all case-sensitive.

Through the terminal server

1. Download the .RUN file to your desktop
2. Open a terminal by going to Application > Accessories > Terminal
3. Type 'CD Desktop' to go to your desktop folder
4. Type 'LS' just to check the .RUN file is in this one
5. Type 'sudo sh nameoffile.run'

Through the Gnome GUI

1. Right-click on the file you downloaded
2. Select 'Properties'
3. Select the 'Permissions' tab
4. Tick the 'Execute' checkbox

Conclusion

The solution using the terminal worked for me but the second one using Gnome didn't.

Apparently everyone knows how to download FLV videos from web addresses which use the Youtube syntax:

http://www.youtube.com/watch?v=