Yes, absolutely, that's how the majority of the tourists do it. The huts wouldn't have had enough space to accommodate even a fraction of the hikers if all of them wanted to sleep there. Personally, I live in Cracow (2 hours drive from Zakopane) and I have no problems with going for one-day hiking trips in Tatras with riding from Cracow in the morning and returning to Cracow the same day, if you're staying in Zakopane you can start the trip much earlier, so it's even easier. I'm reasonably fit but by and large my condition is not exceptional and even not what one would probably call "really good". We even climbed Rysy this way, and Rysy is the mountain which takes the longer to reach in Polish Tatras (it's 12-13km in distance - times two of course, 25km - and 1,5 km of height difference). If took us around 12 hours, if you're accustomed to 12 hours mountain trips and if you can start earlier than us because of staying in Zakopane, you shouldn't have problems, too.
Also, notice than starting from the city you can take much smaller backpack with you, if you go from hut to hut you need all the cloths for several days and the other stuff, and since there are many hikers in Tatras, if you don't book a place in the hut in advance, you have to sleep on the floor, meaning you need the sleeping pad and sleeping bag in your baggage too. I guess it would probably be more demanding for me to hike Tatras from hut to hut with full baggage than it is from Zakopane to the peak to Zakopane with a lighter backpack, though it all depends on the specific person I reckon.
The other idea can be to go from the city to one hut, stay there several days (while going on the trips "light") and then move to another and so on. Staying in the huts is significantly more expensive than in the towns (especially if you arrive by car so you can find an accommodation not near the city center of Zakopane), but even if (I'm sure) you're able to reach every place from Zakopane or other towns, starting from the hut you can maybe do two peaks, two parts of the rigde etc. in one day. You may (or may not) need less refreshment days, too. So the "cost per day" will surely be higher, but the "cost per number of places seen" may or may not be higher, you have to figure out for yourself. It all depends on how many days you intend to spend here I believe.
Two closing points which may be useful:
- Murowaniec hut doesn't admit hikers to sleep on the floor. You have booked a bed in advance (I'm not really accustomed to what the word "advance" would mean here, e.g. if I'll succeed trying to book a bed the same day in the morning or not), or there is one available when you arrive (I wouldn't count on that in Jul-Aug) or they won't invite you.
- Hiking the Tatras is a popular activity in Poland so during summer especially the most famous places like Rysy or Giewont are crowded. Hence, thinking if you can manage a trip in one day (e.g. from Zakopane to Zakopane) you have to take into consideration that the crowd may slow you down compared to how fast you'd walk alone. (It's also a good idea to avoid crowded peaks on weekends, for Giewont likely it's a good enough reason to avoid it at all ;) ).
