I'm guessing that you're passing the entire arrays (ie NaNs and all) into interp1. For cubic interpolation, interp1 calls pchip which excises any NaNs in the data. However, interp1 just performs linear interpolation itself.
For linear interpolation, the interpolated value is simply the weighted average of the points on either side. If either of those is a NaN, the result is a NaN.
Here's some example code to show what's happening, using the data sample you posted:
% Set method to 'linear' or 'cubic'
method = 'linear';
% Data
x = [0.595 0.2243
0.605 0.2421
0.615 NaN
0.625 NaN
0.635 0.2181
0.645 NaN
0.655 0.1911
0.665 0.2479];
t = x(:,1);
y = x(:,2);
% Interpolation, first cutting out NaNs
idx = isnan(y);
y(idx) = interp1(t(~idx),y(~idx),t(idx),method)
subplot(2,1,1)
plot(t,y,'o-')
% Interpolation on raw data
y = x(:,2);
y(idx) = interp1(t,y,t(idx),method)
subplot(2,1,2)
plot(t,y,'o-')
