A good tip I learned: look at your existing tire in the frame and figure out where the point of tightest clearance will be. On the front wheel that's obviously up by the brake and the fork. On the rear, it is probably down by the bottom bracket on the seat stays.
Then, take your allen wrenches and progrssively slide them between that point on the tire and that point of the frame. Start small and work your way up in size, eg: 4mm, 5mm, 6mm, etc. If you can fit 6mm, for instance, on both sides, you have an additional 12mm of clearance. So if you are running a 25, and you can fit an additional 12mm total, you should be able to fit a 28mm tire or even larger.
I aim to have at least 3-4mm of clearance on either side of my tires - to account for a bit of wheel flex and occasional mud.
I also own a set of digital calipers, pretty cheap on amazon, to get an accurate tire size to base my initial measurement on. The printed tire width is often not what the tires actually inflate to, particularly on wider rims (18mm internal width +).