No, stainless steel is INHERENTLY harder than mild steel. It didn't harden up because you tried to drill into it, it was harder to begin with. 304 stainless steel is about twice as hard as mild steel.
If you're going to be drilling into stainless steel, then you're best off buying titanium nitride coated drill bits (the gold coloured ones) and just throwing them away as they dull.
Cobalt bits are harder than High Speed Steel bits, but the titanium nitride coating on titanium bits is the hardest dam thing out there. So, while you may have to spend an extra $10 for three or four titanium bits, you'll save $10 worth of time drilling.
For general all-purpose use, cobalt bits are the best bits to have in your tool box cuz you can sharpen them when they dull and effectively have a new cobalt drill bit. Sharpening a titanium bit grinds off the super hard titanium nitride coated cutting edges on the front of the bit, leaving you with what is effectively an ordinary high speed steel drill bit. So, while you can sharpen cobalt bits for better economy, if the job involves drilling into a hard material like stainless steel, I'd buy some titanium bits and get the job done faster cuz time is valuable too.
Here, read this:
http://www.irwin.com/support-services/ask-irwin/4
PS:: Terry Love:
Could you replace the "Mechanical Engineer" under my UserID with the phrase "Janitorial Technician" if it fits, or "Janitorial Tech." I haven't worked as a professional engineer in over 25 years. Now, I own a small apartment block and MOST of what I do is clean up after people when they vacate their apartments or spill laundry soap all over my laundry room floors or track mud onto my front lobby carpet. Hence "Janitorial Technician" is a more apt job description of what I actually do during normal business hours than Mechanical Engineer. (I also repair, renovate and manage the building, but MOST of my time is spent cleaning up after people.)