Most road bikes built from about the late 90's up until now actually won't go above 25mm, there are few that can go to 28mm. One of the reasons I bought my Lynskey was that it allows for 28mm tire size. It appears though that more and more bikes will in the future allow for 28mm tires but right now the selection is still rather narrow. Even my old steel bikes had a tire limit of about 28mm, but as the years went by and dual pivot brakes bursting onto the market place the brake bridge wasn't as tall as the old single pivots were so the newer brakes can't accept anything wider than 25mm (recently though they have made the bridge a bit taller to accept 28mm tires); also the evolution of CF forks didn't allow for the clearance in the height department for tires wider then 25mm; then you have to worry about the chainstay and BB area, a lot of road bikes have a tight geometry frame and a tire wider than a 25 will rub either the stays and or the BB area. Now a lot of the newest bikes are changing those areas to accept up to a 28, but I remember as late as 2013 when I bought my Lynskey there were very few bikes that would accept a 28 and I was surprised that the Lynskey would.
Also keep in mind that not all tires are exactly as they claim their width is, so one brand of 28mm may fit but another may not, I've even seen some cyclists have issues that some 25's didn't fit but others did. And there were some bikes made that wouldn't accept anything wider than a 23!
If your bike won't go above 25 but you want a low cost CX bike you could buy a vintage touring bike, change to 700c rims since most of those came with 27" rims (but make sure the brake pads will reach the rim), and those will easily fit a 32mm tire and have room left over for fenders if you want, or go with a wider tire and forego the fenders. Or keep the price down and keep the 27" rims on the vintage touring bike and get a set of Conti Ride Tour tire, or the Kenda K161 KrossCyclo which both (especially the last one) has a CX type of tread pattern. The Kenda tires do not have the best puncture belt in them but you could use a liner to beef them up.
