We switched to T-Mobile from Verizon about a year ago and are so glad we did. Our phones now work seamlessly overseas. We both have phones we got through Verizon and before this, we were going through the hassle of getting SIM cards when overseas. Although this can be done relatively cheaply ($20 SIM cards for 30 days of limited data), we were constantly running into issues with multi-country itineraries. Even though we have "unlocked" phones, when we purchased a SIM card that was supposed to work throughout the EU, we were finding that once we went to the second country, the SIM card would stop working. We'd end up spending vacation time trying to trouble shoot the problem (you could not edit the Access Point Name under Mobile Networks, which the overseas carriers said was a Verizon issue). Inevitably, we'd need to purchase a new SIM card in the next country, which turned out to be problematic if you were driving (GPS would stop working) or taking the train (SIM cards not as readily available).
Verizon's international plan is a joke. Even though we were pretty happy with their domestic plan, we finally said "enough" and switched to T-Mobile. Have not regretted it. It is even cheaper than what we were paying for Verizon, and we get international coverage and unlimited data. We used to have to manage the data among 5 family members with Verizon.