from the HSA for America FAQ
If you do not keep your coverage for at least 12 months, the
contribution limit will be pro-rated based on the number of months in
which your HSA-eligible health insurance was in effect. For example,
if your coverage begins on May 1 and is in effect only through
December, you will only be allowed to deposit 8/12 of the annual
contribution limit.
However, if you are eligible to contribute to an HSA on the first day of the last month of the tax year(Dec 1st), you may make a full contribution for the year. However, if you lose eligibility while you are still in the test period(the end of the next year), you will be required to pay taxes on the extra amount, along with a %10 penalty
IRS Pub 969
Last-month rule. Under the last-month rule, if you are an eligible
individual on the first day of the last month of your tax year
(December 1 for most taxpayers), you are considered an eligible
individual for the entire year. You are treated as having the same
HDHP coverage for the entire year as you had on the first day of the
last month.
Testing period. If contributions were made to your HSA based on you
being an eligible individual for the entire year under the last-month
rule, you must remain an eligible individual during the testing
period. For the last-month rule, the testing period begins with the
last month of your tax year and ends on the last day of the 12th month
following that month. For example, December 1, 2013, through December
31, 2014.
If you fail to remain an eligible individual during the testing
period, other than because of death or becoming disabled, you will
have to include in income the total contributions made to your HSA
that would not have been made except for the last-month rule. You
include this amount in your income in the year in which you fail to be
an eligible individual. This amount is also subject to a 10%
additional tax. The income and additional tax are shown on Form 8889,
Part III
Jane(assuming she went straight from one health plan to the next) has a contribution limit of 8/12 of $3300, or $2200. She has already exceeded this, so her company must be assuming she is taking advantage of the last month rule. Under that rule, she can contribute $800 more before April 15th.
John, assuming the same, and that the HSA is with the new company, has a contribution limit of 3/12 of $3300 or $825. If he wishes to use the last month rule, he may make contributions up to the $3300 limit, but he must stay eligible for an HSA for the following year.