You can do it to a degree but browsers will vary where text length varies between browsers and systems especially in fixed width layouts. Assuming you have taken care of all the margin/padding defaults and coded the thing correctly in the first place then often there is little you can do for browsers differences apart from hacking each browser which soon becomes a nightmare to maintain. Older versions of IE always needed this approach but thankfully we are getting past that point now.
I'm not a designer so I always work from a finished PSD usually handed to me at the last minute for coding so I can't use the browser first approach. I code 3 or 4 psd template conversions a week (usually basic fixed width corporate sites) and I manage to get them more or less exact allowing for differences already mentioned. The way I check that it matches with the PSD is to take a screenshot of the browser window and to overlay the section on the PSD to make sure it matches. You can easily see where adjustments need to be made that way. You can even use a background image of the design in the browser window as a guide for testing or [something like this (which unfortunately is [URL="https://addons.mozilla.org/en-US/firefox/addon/pixel-perfect/reviews/?src=api"]not working](http://pixelperfectplugin.com/) in the latest Firefox/Firebug).
However, the important point is to know when designs won't work and what is actually feasible. Margins and padding are pretty easy to get right especially if the design is consistent but details where the designer just stretches text in photoshop to fit or just adds a few more words to make things start on the same line are not feasible. You should be able to spot these flaws from the start and advise the client beforehand.
Pixel perfection is not possible but you can get pretty close so as not to worry about it. I do have clients who want every pixel accounted for but the skill is knowing when this is possible and when it isn't. If you have content on the same line and each piece of text is 2 or 3 pixels out it looks really bad and should be fixed. Remember also that browsers have rounding errors and the odd pixel will always escape you no matter what you do.
Having said all that it seems the landscape is changing and designs need to work on different devices these days so it may be that the days of standard psd to html are numbered as clients request layouts that can adapt to various devices automatically. The clients will still want pixel perfection but you will need to know when that is possible and when it is not.