I believe the issue is in your feature collection. I can't access it as you don't have it shared, so can't investigate what is going on there.
But I have tested the script with a random geometry placed on the map, and had no issues.
Check that your feature collection doesn't have too many features to it.
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EDIT
Thanks for including the link. The problem lies with the fact that you have 1918 features in your collection, and 184 images in that collection.
Multiply those 2 numbers together, and you get more than 5000 elements.
In any case I'm not sure how much information you would extract from a graph that had 1918 NDVI series on it.
Here is the modified code so you can see the print in the console.
var table = ee.FeatureCollection("users/spatola/MCD45A1_broad_leaved_311/MCD45A1_2004_clc311");
print("Original",table);
var subset = table.limit(2);
print("Subset",subset);
var broad_leaved= subset //ee.FeatureCollection('users/spatola/MCD45A1_broad_leaved_311/MCD45A1_2004_clc311');
var modis = ee.ImageCollection('MODIS/006/MOD13Q1');
var modisNDVI = modis.filterBounds(broad_leaved)
.filterDate('2002-01-01', '2009-12-31')
.select('NDVI');
print(modisNDVI);
var timeseries = ui.Chart.image.seriesByRegion(
modisNDVI, broad_leaved , ee.Reducer.mean(), 'NDVI', 250)
.setOptions({
title : 'vegetation series',
xAxis : {title : 'YEAR'},
yAxis : {title : 'NDVI'}
});
print(timeseries);
This code gives the following image.
Perhaps what you meant to do is have a chart for each polygon? That would give you 1918 charts, and I'm unsure how to build it so that you could print it to the console.
.limit(5000)
to plot a subset or export the result to e.g. ExcelExport.table.toDrive()
and plot it there.