You most certainly can but you should do your homework first.
One site, www.cruisecritic.com will give you a plethora of info on ports. You might also team up w/others to share a cab (negotiate the day rate first)to cut down on expenses.

Many cruisers prefer to do excursions on their own. You are in no way obliged to tour with the ships excursions.

By all means do it on your own. Chances are that you will be a lot happier. It does require some research - on the web, cruise sites, etc. Also ask friends who have been to the same places. Look for one that does not offer shopping opportunities because that usually means half the tour in a t shirt shop.

In almost all cases, you can arrange excursions on your own for a cheaper rate than you can get them from the ship. This is especially true in Alaska and Europe. It depends on where you're going, of course, and you can sometimes get a private excursion for about the same price through the ship (a private driver/guide) that you can get on your own. The only advantage of booking through the ship is that it can go on your ship bill and be paid in one lump payment at the end. But I think a lot of cruisers in Europe overdo it with the excursions, which are expensive. Look to the ship if you want to do something that's far away (Paris from Le Havre) or special that you can't arrange on your own (a visit to a particular winery). Do most everything else on your own.

We publish a guide to European Ports of Call as do other publishers. You can make good use of the information in those books to help you do your own touring.

You need to do research on the ports and see if an excursion on your own is worth the hassle. The bad thing about cruise line excursions is the price, but the good thing is that you will not miss your ship if something goes wrong. I plan to book many Carnival excursions on our Mediterranean cruise in early May because in many of our ports, the sights are located quite a distance away. Check out Rick Steve's Mediterranean Cruise Ports and he has a cheat sheet for ports and whether to take the ship excursion or book on your own on his website.

You just have to make sure you get back on board on time. If you miss the boat, you missed the boat and will be on your own. If you book ship's excursion, they will guarantee that they will wait or somehow get you on board. If you venture off on your own, you are on your own. Just be aware of the risk. With that said, we go off on our own and not use the ship's excursions all the time.

I have not booked a ship's excursion in years. We were in Europe this past fall and took private excursions at each port of call. The majority of the private excursion companies/guides know that if they do not get you back to the ship in time, they will be out of business. Reliable guides can be found on the cruise critic website (under ports of call). I have used these guides in Europe, Mexico, the Caribbean and New Zealand. Never have I been late for the ship.

An addendum: www.cruiseexcursions.com. I have not used this company. I tested one port I was at recently & while I paid $53 directly, they are charging about $69 so there is a slight mark-up.

There are some ports where the ships can't dock and you are taken in by tender. I don't believe there is ever a charge for this. Tenders are used for Mykonos and Santorini, for example.

We are doing an Eastern Med cruise in late May and are using ship excursions in Istanbul, a local tour operator from Naples to Pompeii and touring on our own in Athens, the islands mentioned and Ephesus.