My 2013 Cannondale CAAD 10 started developing shifting problems about a week after I brought it home (~2 months ago). It's 105 10-speed. I've taken it back to the shop for adjustment and I've adjusted it but nothing seems to help.
Here's the problem: When shifting from cog 5 to 6 or 6 to 7 whether I'm on the large chainring or the small the chain hangs up and I have to shift again. Then the chain jumps 2 gears. Same when shifting from cog 7 to 6 or 6 to 5.
The bike is brand new. It's a leftover 2013. I've inspected the chain and the cassette.
Got any ideas?

Funnily enough, I had a similar problem with a new 105 10-speed groupset last year. Mine's a triple, but that probably isn't relevant because the issue, like yours, happened independently of which chainring I was on.

Eventually I started from scratch, reinstalling the rear shifter cable and keeping the cable tension at the absolute minimum required to allow me to shift at all. Problem solved. Still have no idea why that worked, but it did. YMMV, of course.

I just adjusted my 9 speed w 105's, here is what I did.
Go to large ring, select the cog in back that puts chain directly in line w cog/ring (no angle on chain), use barrel nut and adjust the rear derailuer until the chain is flowing straight from bottom spinner to top spinner to cog, and hitting dead center chain to cog centered between next cog up and down. (It is very fine tuning- small adjustments).

Lever up to top cog and adjust thee front derailuer high limit just to the point the inside of front derailuer cage clears the chain with rear cog in high gear.

Lever to small cog and make sure outside of cage on front derailuer is not rubbing chain.

Go to small ring and verify front derailuer not rubbing at ether end of spectrum.

I also noted if going to next higher gear not to inadvertently partially press the down shift lever while pressing pushing upshift lever, that seems to result in a shift and a half which resulted in the condition you described.

It now shifts smoothly on stand, I'll see road results tomorrow.

I also have a new bike with a 10-Speed 105 group (5700 I think?). I have a very similar issue. If I'm not extremely precise while shifting up to a larger rear cog, the derailleur will "hang" partially between shifts. If I shift again, it will move up 2 cogs. It's annoying. The 9 speed tiagra group on my last bike never did this.

I believe it's an issue with cable friction. I've heard the internal cable routing on these 5700 shifters can be finicky and if it's not "just right" there can be too much friction near where the cable exits the shifter. I was just getting online to research this some more and to see what's involved in checking the cable routing... (Do I have to remove bar tape? Etc...)

Hum if I still experience issues tomorrow I will examine cable friction

I just returned a brand new Cannondale Synapse Tiagra over similar shifting issues. I couldn't figure it out, and neither could the bike shop. I felt it was internal friction, and I tried everything. They determined that the derailleur hanger was out of alignment. They adjusted it, but it would not hold. After ten minutes, it was back out of alignment. They wanted to order a new hanger, but I had had enough. I exchanged the bike for a Giant Defy 1, which works as it should. Have you LBS check the derailleur hanger.

I have 105 derailleurs and shifters. It has never shifted real good going to the bigger cogs on the cassette. The smaller cogs it shifts good. I backed out the B tension screw a very little bit and that seemed to help. Check out the internet on adjusting the b tension screw and it tells you step by step.

When you say, "adjusted," what adjustments have you made? I'm asking because the easiest change is barrel adjusters and cable tension, going slowly and checking the results by shifting a few times.

Does it work on the stand but not on the road?

I have the 2009 version of 105 10sp, cables coming out the side, not the newer aero style w/hidden cables, so I think mine are the 5600? However, after installing new cables/housings, my middle cogs were funky, sometimes shifting, sometimes not, and then jumping 2 cogs when I shifted again, but only on the road, so sounds similar.

If the problem is cable tension, and if the problem is smaller to larger cogs, then increase cable tension slightly using barrel adjusters, 1/4 to 1/2 turn at a time until the shifting changes, hopefully for the better. One way to prove / disprove is to loosen the tension with the barrel adjusters until it gets worse, then bringing the tension up again.

Road test passed, not a single missed shift, BTW prior to adjustment it was doing exactly as OP described