Amy:
The above replies to your post all contain very useful information and the posters are all well experienced, long-time forum members. Save them, they are valuable.
However, one additional factor to be aware of as you plan your dates, is that St. Jean Pied de Port is in the foot-hills of the Pyrenees Mountains. You will be walking through a mountain pass on the Route Napoleon to climb over the Pyrenees into Spain.
In March and throughout all of April it is not uncommon for the Route Napoleon to be closed by the local authorities due to snow, ice, or poor visibility (fog). People have injured themselves - and worse - over the years by attempting to slog through snow, ice and fog.
There is an alternative Route Valcarlos, that sort of goes around the highest peaks but follows a road. Most folks prefer the over-mountain Route Napoleon (I do). But some prefer the less steep Route Valcarlos. ("Non potest esse de rebus dissensio gustus.")
My point is that, the earlier in the season you go, the more likely you are to be told at the Pilgrim Office at St.Jean that the Route Napoleon is "closed tomorrow," or for "x" days. There have even been posts here and in blogs about people having to take a bus or taxi around the mountains to Roncesvalles to avoid the bad weather off the road when BOTH Routes Napoleon and Valcarlos are closed. It does happen occasionally.
I like to start in the last week of April, after Easter. Many "holiday travelers" will start on the long Easter weekend to leverage the additional "free" days off. But, after the first several days, traffic thins out along the way. So, I give the main peloton a "head start" and trail after them. Hence, I hope to arrive at St. Jean on the Friday after Easter, and start walking on the Sunday after Easter.
Last year I started on 24 April, and there was still ankle-deep snow in parts of the Route Napoleon on the Spanish side of the border, but not enough to be a hazard. The sun was warm and the views were INCREDIBLE! This year as Easter comes later, I plan to start on Sunday 27 April. But I must still contend with the possibility of late season snow.
On the other hand, on 16 May last year, at Cruz de Ferro, I was over three weeks into my walk and well along the
Camino Frances. At the cross, there was about 3 inches (@ 8 cm) of wet snow at the peak / cross and it was snowing steadily. However, one day before, and the following day it was dry and sunny...go figure!
Also, in this regard, many albergues do not open for the season before 01 April. Some of the major places like the large Dutch confraternity run former seminario at Roncesvalles are open year-round. However, some of the smaller and private albergues may not open until 01 April. You need to check ahead. It is not a good time of year to be forced to sleep rough.
I hope this helps.