Ditto to the other two responses. Moved from an H5 myself, to the A300. Yes, the potential for better photos in some areas is certainly there - high-speed motion, high ISO shooting for handheld low light, and the flexibility to shoot outside the fixed lens range of the H5. And in other areas, a DSLR will likely equal the output of a superzoom (in areas where superzooms perform admirably already, such as long telephoto, landscape, daylight, etc.).
And you'll likely find a DSLR will require a bit more attention to your settings, to the technique, and sometimes a bit more tweaking in post to get the most out of the photos (many folks don't realize it, but the photos they love from their P&S cameras are usually quite a bit more 'punchy' and colorful than real life, while most DSLRs are tuned very conservatively to return realistic saturation, color, and contrast...which means the photo may not look like what you were used to from the P&S and initially seem disappointing).
Also, keep in mind that to get anything close to the zoom range of an ultra-zoom, you have to spend comfortably into 4-figures just for lenses - even higher if you are trying to equal the sensitivity level of those ultrazoom lenses (usually F2.8-4.0 or so). Ultra-zooms are great bargains and amazingly flexible...and still probably the perfect camera for 75% of the photo-taking amateur public. I dare say many who have DSLRs would probably get better results, more consistently, and with much less effort, had they just gotten an ultrazoom.
For those who have been shooting with an ultrazoom for a while, and find themselves bumping into the limitations, or wanting to experiment and expand their photography beyond the constraints of the camera...plus are willing to accept the much higher cost and much lower convenience (alot more size and weight for a DSLR and several lenses)...then a DSLR is the next logical evolution.
I personally love my decision. I did start to find myself wanting to push into types of photography beyond the capabilities of my H5. Hand-held low light and fast-shutter night photography, indoor photography, and action photography with fast-moving subjects were the areas the H5 couldn't venture or severely limited me. I made the move to the A300, and committed to buying at least 3 types of lenses - one all-purpose walk-around lens to replace the one-piece convenience of the H5 (the Sony 18-250, which gives me 27mm to 375mm of range, compared to the H5's 36mm to 432mm); one low light lens (Minolta 50mm F1.7), and one big telephoto for wildlife to equal what the H5 could do with the HD1758 1.7x teleextender (Tamron 200-500 gives me 300mm to 750mm, compared to the H5 + 1758 range of 720mm).
My total cost so far has run just about $2,100, which I considered acceptable to pursue and expand my photography...and of course, I'm always keeping an eye out for a few extra specialty lenses to suit other desires (lens lust is a disease you cannot avoid once you get a DSLR).