Earlier this year I ran into problems buying tires for my bike and ended up buying a set that my lbs had in stock. The only downside is they are 25mm and are rated to be inflated to 110psi. The ride is harsh. They are Vittorio Rubino Pro tires so they aren’t bad tires.
I have been thinking about getting wider tires
my bike is a 2013 Specialized Roubaix Comp. It has the original DT Swiss 3.0 rims. Rim brakes.
I have had 28mm tires on it with no problems. I am thinking about getting 30mm tires. From searches on the web it looks like they will fit.
Most of my riding will be on a paved mup or street but if I find a gravel track I might be tempted.
Has anyone put 30mm tires on this model bike? Any reason to not try?

advertised tire width is not always the Actual width, and the rim internal width figures into it too...
your research is solid.
the SL8 model can handle a 40, but the older models are limited to 30s.

"Rated to be inflated to 100 psi" - if that means the maximum-pressure rating, the ideal pressure for your body weight plus bike weight might be somewhat lower.

Vittoria Tire Pressure Calculator

ride at a lower psi
what psi did you put into the 28mm tyres
2mm more won't make a difference

I usually like to inflate my tires to as low a pressure as possible without running the risk of rim damage from the many potholes and craters where I ride. This is around 90 psi front and 100 psi rear for my current 700x25 Continental GP 5000's and can vary slightly for other tires. I have tried going lower but it's just too risky. As has been mentioned, if you want more comfort, lower pressure will help but there is always a limit to how low you can go.

I'm surprised at how many inflate their tires to maximum pressure rating. I'm on the thin side, and my bike isn't a heavy one. If I inflated anywhere close to max, it would be like riding a solid tire. Mine are tubed, and for a decent ride, they're from 20 to 30 lbs. lower than max.
Wider tires may provide a more comfortable ride. Try lowering the pressure first, you might find the tires you have are ok.

I think I was running 25s at about 90psi max. I weigh 82kgs.

I weigh 250lbs, or about 113kg.

I’m not thin- 250lbs. It does feel like a solid tire. I ride with tubed tires.
The pressure range on the sidewalls is higher than most other tires.

PB ... I don't want to appear insensitive but, it needs to be said, you don't have a 'comfort bike'. I'm not actually certain that at your weight you can get away with using less than the max pressure in a 25 or 28. Don't fall for the claptrap of the tubeless contingent. 'Comfort' and 'racebike' are mutually exclusive concepts. It shouldn't be injurious, or actually painful, but you are not going to discern much if any difference in comfort between the OEM 25mm and 28mm, as you already discovered. 30mm isn't going to cross some threshold either. HTFU and ride that beautiful bike of yours.

Rubino Pro's were my first venture into a better tire than just the cheapo sub 20 dollar a pop tire I use to use. And they were way more comfortable riding at over 110 psi than the 70 psi I put in the Continental Ultra Sports I was using prior. And those were both 23mm wide tires. Also, that was about 15 years or more ago, and the current versions of those tires probably don't resemble my experience with them.

So perhaps you are looking for a comfort level that just isn't there for your ride conditions. I have extremely smooth asphalt where I ride. Both on the trail and on the road. Though to look at them you'd think they were bad. However I steer around the bad parts. And for stuff that can't be avoided, I get my butt off of the saddle. Even if just a reduction in how much weight I'm putting on the saddle at that moment.

Similar with my hands too. I also loosely grip the handlebars so all that vibration from the bumps doesn't go directly into me. Once you learn how to do both your hands and butt, you'll be surprised at how natural it becomes anytime you see a bumpy section coming up. Even on downhill twisty turns where the pucker factor is high.

Though 110 psi probably is high. You should try much less and see if that also makes for a uncomfortable ride. 70 or 75 psi wouldn't be a unreasonable start. After several rides, if the ride feels better comfort wise, but it wears your legs out, then increase the pressure. Maybe by 5 psi. And then ride several times again to see what you think of that. Rinse and repeat as is necessary. Don't forget to check what your actual stats say for your performance. And pinch flats are generally a indicator that you needed more air pressure.

As for how big you can go with the tire width, just measure the place on the bike that is closest to the tread or sidewall. Tires of the same model tend to grow in height by about the same as their change in width. Typical interference will be the seat tube, the brake bridge if rim brakes or the rim brake calipers themselves. Or the underside of the fork crown.

Also remember that stated tire width isn't necessarily the actual tire width if you are changing models of tire.

Check the clearance for yourself. Slip a 5mm allen wrench into the gap between the tire and the frame/fork and see if it will go.

I am in no hurry. These are good tires. When they need to be replaced I will get either 28mm or 30mm.