Exactly. But it's obviously not just the accent thats at issue here. If you think about it, and try to do a good impersonation of someone talking in a different accent it has to include a bit of dialect and local grammatical idioms to make it convincing.
I assumed that the original comment about 'foreign accents' also encompassed the grammatical tics that foreign speakers often have. Someone speaking grammatically correct english in a strong accent is easily understood, but it only takes a very little grammatical mistake to change to meaning of a sentence. For example, an Italian colleague who missed out a 'to' and told me 'just a minute, I'll come over your desk' caused some admittedly slightly childish giggling in the office...
I assumed that the original comment about 'foreign accents' also encompassed the grammatical tics that foreign speakers often have. Someone speaking grammatically correct english in a strong accent is easily understood, but it only takes a very little grammatical mistake to change to meaning of a sentence. For example, an Italian colleague who missed out a 'to' and told me 'just a minute, I'll come over your desk' caused some admittedly slightly childish giggling in the office...